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An elementary manual of statistics,

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MODERN COMMERCIAL TEXT-BOOKS

FOR USE IN SCHOOLS AND COLLEGES
AND FOR MEN OF BUSINESS

Edited BY LAWRENCE R. DICKSEE, M.COM., F.C.A.

Formerly Professor of Accounting at the University cf Birmingham ^
noiv at the London School of Economics (University of London^

AN ELEMENTARY MANUAL
OF STATISTICS

ELEMENTS of STATISTICS

A. L. BOWLEY, M.A., F.S.S.

READER IN STATISTICS IN THE IINIVERSITY
VF LONDliN

Demy 8vo. Cloth. Numerous Diagrams.
lOs. 6d. net.

This is a more advanced book than the Author's
" Elementary Manual of Statistics," and is intended
for students who possess sufficient knowledge to
follow the processes of mathematical analysis applied
to statistics.

P S KINfi & SON ORCHARD HOUSE
r. J. IVinu W OUl^, WESTMINSTER.

AN ELEMENTARY MANUAL
OF STATISTICS

ARTHUR L. BOWLEY, M.A., F.S.S.

READER IN STATISTICS IN THE UNIVERSITY OF LONDON
AUTHOR OF "ELEMENTS OF STATISTICS," "NATIONAL PROGRESS SINCE 1 88 2,'' E

LONDON

MACDONALD AND EVANS

4, Adam Street, Adelphi, W.C.

1910

BiOHAED Clay & Sons, Limited,

BREAD STREET HILL, K.C., AND
BUNGAY, SUFFOLK.

PREFACE

This manual is intended for the use of those who desire some
knowledge of statistical methods and statistical results without
going deeply into technicalities or undertaking mathematical
analysis ; it is hoped that it will be of service to all who have
occasion to use statistics in their own business or profession, or
who take an intelligent interest in public affairs.

It is also designed as a first course in statistics for students who
wish to proceed further in the subject and, if it serves its purpose
will stimulate interest in the many fascinating problems that
await solution, and that can only be attacked by the methods of
modern mathematical statistics.

The first part deals with elementary methods and with such
technical terms and ideas as are indispensable in the handling of
numbers on a large scale. In the second part the origin of many
groups of public statistics is shown, their adequacy is criticized,
and some of the more interesting results which are based on them
are briefly summarized. This part is intended as a guide to official
statistics, riot as a compendium or dictionary of them ; and the
problems attacked are given rather as illustrations than as sub-
stantial contributions to knowledge.

To facilitate the use of the book in the hands of teachers a
number of exercises of various degrees of complexity are given in
Appendix I. Every serious student of commercial or public
affairs should be acquainted with the nature of the contents of
the Statistical Abstract of the United Kingdom ; to promote this
knowledge, and because many pages of headlines and figures
would otherwise have been necessary, a large proportion of the
examples relate to tables in the Abstract for 1909, for which future
or earlier abstracts can readily be substituted.

Appendix II contains a short list of Blue Books which should
be easily accessible in the Library of every institution where the
subject of statistics has a place in the curriculum.

My thanks are due to Dr. Dudfield (Medical Officer of Health
for Paddington) and Professor Cannan for most useful criticism of
some of the chapters, and to Mr. G. W. Palmer for help and advice
in the correction of proofs. I shall be grateful for any criticisms
which will tend to increase the utility or improve the accuracy of
the book.

A. L. B.
Beading,
December 1909.

CONTENTS

PART I

CHAP. PAGE

I. NATUEE AND USB OF STATISTICS . . 1

II. ACCUEACY AND APPEOXIMATION ... .6

III. AVERAGES 15

IV. THE ACCURACY OP AVERAGING AND OTHER ARITHMETICAL

PROCESSES . ...... 27

V. USE OF DIAGRAMS . . 35

VI. TABULATION . 50

VII. SAMPLING . . . ... 56

VIII. EULES FOE USING PUBLISHED STATISTICS . . 64

IX. METHODS OP STATISTICAL ANALYSIS . . .71

PAET II

I. THE POPULATION CENSUS ... 85

II. VITAL STATISTICS . . .102

III. TEADE AND TRANSPORT . Ill

IV. PRICES . . . .126
V. PRODUCTION ..... 133

VL VfAGES ... . . 137

VII. EMPLOYMENT .151

VIII. OTHER WORKING-CLASS STATISTICS .... 161

IX. INCOME AND CAPITAL . . . . .171

X. TAXES AND RATES 185

APPENDIX I. EXEECISES 197

XI. SELECTED LIST OF BOOKS OP EEFEEENCE . 211
INDEX 213

PART 1

COEEIGENDA.

On p. 155, 7th line from bottom, for only once {1904) read

On p. 156, year 1904, Column B^, for 7-6 read 5-0.
On p. 160, line 1, read is little above the percentage 5-0.

SECOND EDITION

CHAP. XVIII., Consisting of 30 pp., is devoted to an ex-
position of the value and use of Index Numbers.

A PLAIN GUIDE TO
INVESTMENT AND FINANCE

By T, E. YOUNG, B,A., FJ.A*

Past President of the Institute of Actuaries, Past Chairman of the
Life Offices' Association, late Head Actuar:^ of the Commercial
Union Assurance Co*

Crown Svo, 350 pp^ cloth* Price 5s.

A few of the subjects dealt with : — Trade, Commerce and Industry — Mar-
kets generally and the Money Market in particular — Bank of England —
Joint Stock Banks — Bills of Exchange — Foreign Exchanges — Bill Brokers —
Stock Exchange — Brokers and Jobbers — Bulls and Bears — Sympathy of
Markets— Causes of the Variations in the Prices of Securities — Effect of "War
on the Prices of Securities — Dear and Cheap Money — Condition of Trade in
its Effect on the Prices of Securities and Consols — Accrued Interest as affecting
the Cost of Securities — The Return derived from an Investment — Ex-Dividend
and Ex-Interest, Cum Dividend and Cum Interest — Sinking Funds, Specula-
tions and Gambling — The Phase of Commercial Crises in the Com-se of Trade,
and the Succession of Sun Spots — Index Numbers, etc. , etc.

" It is carefully and lucidly written, and any one who desires to get a compreliensive
grasp of the financial world at large and of the Stock and Money Markets particularly,
cannot secure a better or more reliable guide." — Financial Times, Jan. 4th, 1909.

"Regarded from, every point of view this work is really admirable, and the reader
cannot fail to profit very greatly from its clear exposition." — The Financier.

"Mr. Young's book truly fulfils the office of guide. Everything that is necessary for
one to know is expounded clearly and concisely, and the various influences to which the
investment market is subject are shown. The book is a veritable vade mecum and all
Interested in finance and financial operations should not be without it." —

The Financial Standard.

" We are glad to testify with emphasis to the excellence of this manual. It is one of
the soundest, moat carefully written, honest, and lucid manuals on the subiect dealt with
we have ever come across." — TJie Investor's Review.

" This is a book which would amply merit more than one notice. No other author is
better equipped than Mr. Young to supply a real and wide knowledge. Nothing super-
ficial will ever be found in his pages. ... To sum up, we give a most cordial welcome to
a very valuable book. The man who relies on his experience will learn much from its
exposition of principles, and the student who resorts to it to equip himself in principle
will find abundant assistance to him in the at first bewildering world of practice."

— Post Magazine and Insurance Monitor.

" The style is invariably clear and interesting ; and as the book is the outcome of long
experience, it should be found of service both by those who have money to invest and by
the ordinary student of economics."— Tft,e Scotsman (Edinburgh).

. "We must highly commend Mr. T. E. Young's *A Plain Guide to Investment and
Finance' for its sound reasoning, solid good sense, and wise discrimination."

— The Liverpool Post.

" It is a remarkable book in many ways, and is thoroughly worth the price asked. It
should find a place on every business man's desk, for there is certain to be something in
it that can be learned with advantage*" — The Review.

"Mr. Young's name is familiar to everybody in the life assurance world, where he has
made a reputation for sound judgment and an absolute mastery of his profession. . . .
From start to finish his book is couched in simple language, with explanations of the
few technical terms in footnotes. . . . It is excellently written."— r/ie PoZtcy.

MACDONALD & EVANS, 4 ADAM ST., ADELPHI, LONDON, W.C.

AN ELEMENTARY
MANUAL OF STATISTICS

CHAPTER I

NATURE AND USE OF STATISTICS

1. Statistics are numerical statements of facts in any
department of inquiry, placed in relation to each other;
statistical methods are devices for abbreviating and classify-
ing the statements and making clear the relations. The
elementary methods are based on arithmetical processes
of an easy but specialized kind ; more refined methods,
necessary for certain classes of investigation, involve complex
mathematical ideas.

2. Statistical treatment is necessary in a very great variety
of cases, some of which may be distinguished as follows —

Groups. — If a large number of things or persons have
something in common, e. g. as members of the same nation,
workers in the same occupation, houses in a defined locality,
but dififer one from the other in respect to some measurable
characteristic, e.g. age, amount of wages, rateable value,
together they form a statistical group. Groups can be repre-
sented by diagrams, tahdated in grades, or described in
abbreviated form by averages.

Classes. — 'If the characteristics in which the things or per-
sons differ are not measurable, but need separate description,
e.g. the number of persons in different districts, or in different
occupations in the same industry, or of houses used for
B 1

2 AN ELEMENTAEY MANUAL OF STATISTICS

different purposes, a statistical table can be made showing
in juxtaposition the numbers in various classes and sections
according to any scheme of classification, and the relative
sizes of the classes can be indicated by percentages.

Series. — If the numbers in some group or class are counted,
or the quantities or values of some aggregate are measured,
periodically (weekly, monthly or annually), we obtain a statis-
tical series, whose nature is most easily appreciated with the
help of a diagram.

3. Statistics are thus used for describing and analyzing large
groups or aggregates, too large or complex to be intelligible
by simple observation. Thus the affairs of a community, the
progress of a large business, and the productivity of a country
need statistical treatment, while the individual, the single
transaction, the quantity grown in a field do not. The differ-
ence is not one of degree only, for when investigation is
extended over a large area, regularity is obtained, conformity
to general laws is visible, and new methods of description are
required, while observation of a few cases suggests only
chance and chaos. There is infinite variety in the constitu-
tion of a family, but in a community the distribution by age
is nearly invariable. Men differ from each other in stature
and in wealth; but simple mathematical formulae describe
the distribution as to height and as to income of the members
of a nation. Statistics generalize and repair the defects of
individual experience.

4. Statistics are specially useful for making comparisons of
similar aggregates from time to time, or from place to place.
The significance of one quantity, e.g. the average wage of a
group of workmen, can only be appreciated by comparison
with anothei', e. g. the average wage of another group in a
different occupation or district, or the same group at an
earlier date. The gradual change of the birth-, death- or
marriage-rates during a series of years, shows very much
more than the statements for a single year. Again, it is
frequently necessary to show the relation of one quantity —
for example, the total importation of wheat — to another, for

NATURE AND USE OF STATISTICS 3

example, the population; or, to take another instance, the
relation of the total wheat crop to the area under cultivation.
The choice and exact definition of the aggregates that should
be thus brought in relation to each other are by no means
simple matters.

5. When observations are thus extended, many sources of
inaccuracy are found to be present, and it is very frequently
impossible to remove them completely. Statistical results
are, therefore, very generally estimates rather than exact
statements, and it is a matter of the very greatest import-
ance to learn to what degree of accuracy various statements
can be trusted, and to obtain methods of neutralizing the
effects of errors and omissions of all kinds.

6. Perhaps the principal cause of incorrect use of statistics
is want of attention to the definition, meaning and limitation
of each estimate quoted. A total, such as the population of
England and Wales, or the total value of goods imported
into the United Kingdom, is generally the result of a compli-
cated system of enumeration, in which a large body of per-
sons have co-operated, working under printed instructions.
To know what is included in the total implies not only care-
ful reading of the title, " Total value of Foreign and Colonial
Merchandise Imported," but also knowledge of the method
of valuation, of the definitions of " Merchandise " and of
" Imported," and of the nature of the omissions (goods
brought in as personal luggage or smuggled, etc.). There
is hardly any total whose full meaning is apparent simply
from its description ; there is always to be implied some
such phrase as " so far as the items are included in the
working definition and enumerated by the staff concerned."
The total or average used is a total or average of many items,
each of which satisfies some complex definition ; this defini-
tion is not thoroughly known till the whole method of
collection and tabulation is known. In many cases the
necessary explanations are given in the introduction to or
the footnotes of an official report ; in others, where informa-
tion is not forthcoming, extreme caution is necessary in

B 2

4 AN ELEMENTAEY MANUAL OF STATISTICS

using the figures, till a careful inquiry as to their meaning
can be made. In Part II below, some of the more important
definitions are given.

7. It is frequently the case that the quantity as to which
knowledge is desired is not capable of numerical measure-
ments. We cannot measure health, poverty or crime ; . we
can only measure the death-rate, and count the number of
persons who receive public relief, and the number of convic-
tions. In such cases the measurements can only be used as
indications, and their relation to the more important quantity
must be constantly criticized, while other indications should
be obtained wherever possible to check the impressions
formed. Thus the number of paupers changes with altered
administration, and of criminals with modifications of law,
and the death-rate differs with age, sex, locality and occupa-
tion; but in these cases we have other means of knowing
and testing the changes in the quantities concerned.

^. It is very important to avoid mistaking the part for the
whole. The growth of exports is often used as an indication
of the general growth of trade, but the more important home
trade has not been measured, and the whole may diminish
while exports increase, or vice versa. The number of mem-
bers of certain trade unions out of work is published
monthly, but the percentages based on them cannot be used
to measure unemployment as a whole without many qualifi-
cations. If our definitions are correct they will show the
limitations in extent of our estimates. Other cautions as
to common mistakes in using statistics will be found scattered
through the chapters that follow.

9. Three of the principal uses of statistics are (i) to give
correct views, based on facts, as to what has happened in the
past; how, when and under what circumstances, population,
trade, wealth, etc., have grown; and by comparison and
analysis to search for the causes of changes that have taken
place ; (ii) to afford material for estimates for the present,
e. g. the probable yield of a new tax, the amount of trade that
will be carried by a new route, the quantity of water needed

NATURE AND USE OF STATISTICS 5

by a town ; (iii) to make possible a forecast for the near
future; for this purpose we study the changes that have
taken place in the recent past, by the light of the relations
between phenomena that comparative statistical analysis
reveals.

10. The main sources of statistical information are (i)
official tables published periodically by various Government
Departments, (ii) the results of special inquiries made by the
Departments or by Eoyal Commissions or Parliamentary
Committees, (iii) monthly and annual reports on special trades
made by Chambers of Commerce, trade newspapers and
private firms; (iv) special investigations made by private
individuals as to social conditions. All of these have their
limitations and present special difficulties, and together they
are quite inadequate to afford sufficient information as to
most of the conditions of welfare, progress and trade which
form the subjects of inquiry. There is urgent need for more
systematic and more complete national statistics.

11. Even if statistics were complete and perfect, their use
would be definitely limited to one aspect of a problem, that is,
the numerical aspect. Statistical results are essentia], when
judgment is to be formed on any questions that involve num-
bers, quantities or values, but they must always be brought
into relation with the personal, political, aesthetic or other
non-quantitative considerations that may be of greater im-
portance in deciding on a course of action. Statistics only
furnish a tool, necessary though imperfect, which is danger-
ous in the hands of those who do not know its use and
deficiencies. A knowledge of methods and limitations is
necessary, if only to avoid being misled by unscrupulous or
unscientific arguments.

CHAPTER II

ACCURACY AND APPROXIMATION

1. Perfect accuracy is very seldom obtained in statistics,
and in this respect they differ from accountancy. A state-
ment of fact involving £ s. d. can be made exactly, and must
be so made to afford a perfect balance, but as soon as we
deal with quantities or values, where the things counted are
not perfectly similar to each other or are matters of estimate,
we can no longer give an exact unqualified statement. There
is no means of knowing exactly the quantity of wheat grown
in the United Kingdom, both because one bushel of wheat
differs from another in dryness, fineness, and other respects,
and because the whole bulk is not and cannot be measured,
but is estimated from the acreage under wheat and the average
productivity. We cannot know the population of England
and Wales exactly on June 30, 1909, for it is eight years
since the population was counted ; no record is kept as to the
numbers who have gone to or come from other parts of the
United Kingdom, statistics of emigration to and immigration
from the colonies and foreign countries are imperfect, and
probably a small number of births are unregistered. Both
these totals can, however, be estimated with considerable
accuracy.

2. In such cases we should not say that the population
consists of 35,751,963 persons, or that 56,531,198 bushels
of wheat were produced in 1907, except as bare numerical
results of a calculation ; but we should aim at finding to how
many figures the statements are likely to be correct.

Supposing this difficult operation performed, and that (for
example) an error of 100,000 persons is possible in the

ACCURACY AND APPROXIMATION 7

population and of 250,000 bushels in the estimated pro-
duction of wheat, various methods of statement are open
to us.

{a) The population is 35,751,963 + a number not greater

than 100,000.
(6) The population is 35,750,000 + 100,000, or 3575 + 10
(OOOO's omitted).

(d) The population is 36 X 10«, or 36,000,000, to the

nearest million; or (in a table involving other
similar figures) is 36 (000,000's omitted).

(e) The population is 35,750,000, correct to 'S/, or

to 3%,

(/) The population is 35g^ x 10* (where gl is not a frac-
tion, but an abbreviation for " between 65 and 85 ").

If the error were, however, known to be not more than 2,000,
we could make a shorter statement, viz. that the population
is 35,750,000 "in round numbers " or " correct to 10,000";
for the maximum and minimum possible, viz. 35,753,963 and
35,749,963, are both nearer to 35,75 than to 35,74 or 35,76
(OOOO's omitted). This is the best method when applicable,
but in the case given we cannot be sure which is the nearest
100,000, and {d), which is the corresponding statement,
is unnecessarily rough.

Each of the above statements would be correct for some
purpose ; the choice depends on the nature of the table of
which it is to form part, (c) is the clearest if we are not
making a table, (e), or an equivalent form, is the most
scientific. (/) has not actually come into use, but may be
suggested as the most compact way in which the whole data
can be stated.

3. When round numbers are used, the last digit retained
must be the nearest to the estimate, not the next under.

Thus 374,563 is 374,56" or 3746°'' or 375»<"'* or 370'""', ^^^
3745"" and 374""''. In the third case, the number being

* This is merely a convenient way of writing 375,000, when it is
implied that the number is correctly given only as far as the 5.

8 AN ELEMENTARY MANUAL OP STATISTICS

nearly midway between 374,000 and 375,000, it would be
better to write 374-5 (thousands). Round numbers are
employed both as abbreviations of nearly exact statements and
to indicate the accuracy of estimates, the last digit given
being supposed correct.

4. The arithmetic of inexact numbers needs special at-
tention. Unnecessary work is to be avoided; only those
digits should be given in the result which are supported
by the premises : an indication of the possible error should
be given. The following five examples show various ways in
which the work can be carried out.*

Additimi.— Add 47,38(), 94,53, 843,782, the numbers being
correct to 2 ^, 5 %, and '5°/^ respectively.

Then the first number is only given as between 47,386 +
948 and 47,386 — 948, and the work may be set down as
follows : —

47,386 ± 948

9,453 ± 473

843,782 ± 4,219

900,621 ± 5,640
Answer, 900,000, correct to -6 %, or " between 895,000 and 906,000.

If a less exact answer is sufficient, we may notice that the

last entry makes the greatest contribution to the error, and

write

47 OOO's omitted.
9
84«

90 X 10*

Subtraction. — Subtract £85,460 from £197,000, the
numbers being correct to the last digit (other than 0)
given.

Then the first quantity is only given as between £85,455
and £85,465, the second as between 197,500 and 196,500.

* The concise statement given in these paragraphs will, it is hoped,
be sufficient for capable arithmeticians, and a fuller treatment would
be out of place ; but these or similar methods are to be found in modern
Arithmetics, to which the reader is referred if the ideas are not clear.
In the end every one makes his own rules for abbreviation.

ACCURACY AND APPROXIMATION 9

Work showing
the Maximum difference 197,500 the Minimum difference 196,500

85,455 85,465

£112,045 £111,035

Answer, £lli x 10^, or £111,500, correct to -5 %.

Multiplicadon.—'MultiTply £30 18s. 6d. by 347,100, the
numbers being correct to the nearest 6d. and the nearest 100
respectively.

Greatest possible errors : — 3d. in £31, or 1 in 2480 ; and
60 ia 347,100, or 1 in 7000.

First method.

Product if there were no error.

Maximum product.

£30 -Sf^^
347,100

£.30Sj^X
347,150

92775

92811

12370

12375

2165

2166

31
15

£10.734.1<"1

£10,739,8"''
Maximum error ± 5, 700 or '53, %^, where '53 %„ stands for '53 per mille.
Second method. — Observe that the answer can only be
correct to four significant figures.

The maximum errors in the factors are "40 ^^ and "13 %;,.
Where small percentage errors occur in factors, both being
in excess or both in defect, it is easily shown by algebra or
geometry that the error in the product is the sum of the
errors in the factors. The product is therefore subject to
an error of 'SS/^^.

3-0925 X 103

10,413

312

£10,733 X 103
Answer, £10,733,''«'', correct to -5 %„ or £107^ x 10*.

Third method. — A little experience will show that the

10 AN ELEMENTARY MANUAL OF STATISTICS

more serious error comes from the first term and is roughly
•4%^. The work should then be done to five figures, and
the answer given as doubtful to one unit in the fourth figure.

Z'msiow.— 45,340,000 tons are valued at £74,380,000.
Find the value per ton, the numbers being correct to the last
digit (not 0) stated.

First method. — The maximum error is obtained when the
dividend is greatest and the divisor least, or vice versa.

Maximum possible
45,^^^)74,385(£l-
45,335

value.
6408

Value if there were no error.
45,^^0)74,380(£1'6405
45,340

29,050
27,201

29,040
27,204

1,849
1,813

1,836
1,814

36 22

Answer, £1 12s. 9Jd., to nearest farthing.

Second method. — The maximum errors are 1 in 9,000 and
1 in 15,000; if cumulative they make -11 + -07 = ■18%„,
that is \ of one farthing in £1. The quotient, worked on
the supposition that there is no error, is therefore correct to
the nearest farthing.

Square root. — Find the length of the side of a square field
whose area is 15 a. 3 r. 29 p., correct to a square pole.
Square poles.

2,549(50-488 poles = 277-68 yards.
25

1004)4900
4016

884
The area is correct to 1 in 5000 ; the side can be, there-
fore,* obtained to 1 in 10,000, and may be stated as 277'? J
yards, or 277-7 yards.

* The relative error is doubled by squaring, and, conversely, halved in
taking the square root. For, if x is a quantity subject to a small abso-
lute error ex, x{l ± e) will be the limits of the approximation to the
value of X. Then a!''(l ± ef, which nearly equal a?(l ± 2e), since e^ is
small, will be the limits for the value of x% which is therefore subject
to a relative error 2e.

ACCURACY AND APPROXIMATION 11

5. Multiplication, division and square root can be more
rapidly performed by the use of logarithms, but there is
considerable risk that part of the data will be lost, or a
spurious accuracy introduced. If the data are correct to four
figures, four-figure logarithms should be used, and the answer
may be depended on to at least three figures, and similarly
with other degrees of accuracy. Slide rules can also be used
for special purposes, but their adequacy must be tested.

It is necessary to call attention to the complexity of these
processes, because it is so commonly assumed that they are
not worthy of attention. It is only a very competent arithme-
tician or experienced statistician who can see the effect of
the inaccuracy of data throughout a problem. It is probable
that many published statistics are less accurate than they
appear, simply because the effect on the results of errors in
the factors has not been considered.

It is to be observed that it is the most inaccurate of the
factors or terms that governs the inaccuracy of the result.

6. Few statistical measurements are accurate to five
figures, many not to more than three, and some are doubtful
in the second figure. On the otlier baud, it is seldom that
greater accuracy than 1 in 1,000 is required, and this can
often be obtained.

It results that, in general, much space can be saved in
tabulation and more accuracy be in reality obtained, by
giving numbers only to three or four significant figures.

7. Comparison and ratio. — It is so much the custom to
make comparisons by means of percentages, that the arti-
ficiality and, in some cases, the fallacy of the result are not
perceived.

Suppose that we wish to compare two quantities, e. g. the
aggregate values of Exports of Home Produce in 1898
(£294,014<""') and in 1907 (£517,977''»''), and that we can
depend on these values to four figures.

Any one of the following ratios expresses the facts —

2940 : 5180 = 1 ; 1-762 = "5676 : 1 = 100 : 176-2 = 1000 : 1762
= 56-76 : 100 = 567-6 : 1000 = 100 - 43-24 : 100.

12 AN ELEMENTARY MANUAL OF STATISTICS

The ratio in italics is the simplest of these statements, if
we take the value in 1898 as the standard of comparison,
and that next written ('5676 : 1) if we take 1907 as the
standard.

The statement most usually made would be {a) " The value
has increased 76'2 // " ; it is more exact to say " The value in
1907 was 176-2 % of that in 1898." The converse is "The
value in 1898 was 56-76 % of that in 1907 " ; the equivalent
of this is (6) " The value in 1898 was 43-24 % less than that
in 1907." Few people would recognize that (5) was the
converse of (a).*

After the phrase " per cent.'' the words " of x " are im-
plied, where x is supposed to be known from the context.
But the context does not always give definite information,
as the following example of evidence given to a Royal Com-
mission shows : " Wages were 15s. in 1870 ; they rose 20 %
between 1860 and 1870, and 10% more by 1875; by 1885
wages had fallen 25 %." Any of the following would satisfy
the statement —

1860. 1870. 1875. 1885.

12/6 15/- 16/6 12/4^, reckoning each period by itself.

12/6 15/- 16/3 1.3/l|, reckoning all on the 1860 basis.

12/6 15/- 16/3 12/2|, reckoning the last on 1875.

12/4^ 15/- 16/3^ 13/o|, reckoning all on the 1885 basis.

From other evidence it appears that the third of these lines
was intended.

One of the greatest strikes of the last decade was caused
by a misunderstanding of this kind.

8. It would be an improvement in common methods if the
decimal point were not used in comparisons ; thus the state-
ment as to exports would read : " the values are in the ratio
1000 to 1762." It would be a greater improvement if the
ratio were always given, not the increase ; thus " the value

* If a! and y are two numbers, y is 100 x ^~ "^ = say, «. per cent.

<w grj y^

greater than x, and x is 100 x " = say, v per cent, less than y.

Then the simplest relation between u and v is 100 (u~v) = uv.

ACCURACY AND APPROXIMATION 13

has changed in the ratio of 1000 to 1762," not " has increased
76-2 7."

Apart from the greater definiteness of the ratio statement
we gain a further advantage in preserving the measure of
accuracy. If average weekly wages change from 25s. 2d. to
27s. 3cZ., each quantity being given correctly to the nearest
3A, the ratio is between 25s. lO^d. : 27s. l^d. and 25s. 7Jd :
27s. 4^d, i.e. between 1000 : 1048 and 1000 : 1068, or may
be written 1000 : 1058 + 10, and is known to 1 %. But
the increase is only known as between 4'8 and 68 ^, or as
5'8 + 1"0 ^, and is doubtful to the much greater extent of 1
part in 6. This source of inaccuracy is frequently ignored.

9. There are two groups of cases in which percentages (or
per thousands, etc.) can be used without indefiniteness ; they
can be shown sufficiently by examples —

{a)

Value of Imports, received by the
colonies, etc., from

The United Kingdom
British Possessions .
Foreign Countries

Total. . £3091 100-0 1,000

In a long column of this sort, the percentage items, each
calculated correct to the third figure, will not give in general
1,000 exactly as the total ; the items should, nevertheless, be
left as they are calculated.

(6) The second group is illustrated by the statements : " Per
million males over 10 years of age in 1901 in England and
Wales, 92,811 were occupied in luilding and works of con-
struction, as compared with 34,898 per million in Ireland " ;
" Per thousand persons in England and Wales in 1871 and
1901, 437 and 470 respectively were between the ages 20
and 55."

Such methods of arranging numbers for comparison can
hardly be distinguished from averaging, as dealt with in the
next chapter.

Per cent.

.„ Per mille

of total. "' of total.

OOOOO'S

£1434

46-4

464

561

35-5

355

1096

18-1

181

14 AN ELEMENTARY MANUAL OF STATISTICS

10. The following examples illustrate common mistakes in
the use of percentages —

"Of 57 persons, 35 (or 61-404%) died." The number in
the brackets is an example of spurious accuracy. In dealing
with less than 100, the figure in the unit place is not
established, and the decimals are absurd.

" Exports increased from £1,000 to £1,300, i. e. 30 %, but
imports increased 500%, the values being £20 and £120."
Here are compared relative increases on values which are so
different as not to be comparable ; the absolute increase in the
first case is three times that in the second. Such a state-
ment is numerically correct, but is likely to be misquoted
simply as " Exports increased 30 % and imports 500 %."

" Prices rose 20 % and then fell 20 %, returning to the former
level." If the most natural meaning is given to the first
clause, the three prices would be in the ratio 100 : 120 : 96,
and the last price would be 4 % below the first. This kind
of ambiguity and the resulting mistakes have already been
discussed (p. 12).

" The total rose from about £143,000,000 to £185,473,000,
an increase of 29-7 %." This should be " about 30 %."

CHAPTER III

AVERAGES

1. Averages are of many kinds and have many uses.
Here we deal only with the simpler averages and kindred
quantities in common use, not involving mathematical
analysis; and, avoiding formal definitions, we explain the
methods and ideas by examples.

" 1,000 cattle in the United Kingdom produce on the
average 58 tons of meat per annum." We cannot say
" 1 cattle produces '058 tons," for this is not true of an indi-
vidual ox, cow, or calf ; the use of the generic noun " cattle ''
itself suggests the more general statement.

An average of this kind is obtained by estimating the
number of cattle and the amount of meat produced year by
year over a period of years, and dividing the amount by the
number.

The use of the statement is partly to abbreviate and to
state in an accurate form (see last chapter) the result of a
complicated investigation ; partly to afford a basis by which
the yield of the herds of the United Kingdom in future years
can be estimated ; * partly to make a standard of comparison
with other countries and other dates.

2. In the census of 1901, 32,527,843 persons were enume-
rated in England and Wales, the area being 37,327,479
acres. There were, therefore, 0-871 persons per acre. In
Worcestershire the "density" was 1'13, and in the county of
London 60*62 persons per acre.

* The number of cattle is estimated by the Board of Agriculture
every year ; the quantity of meat is not estimated officially at all. — See
atatistical Jawrnal, 1909, p. 316.

16 AN ELEMENTARY MANUAL OF STATISTICS

This is an example of a fictitious average. To realize it,
we have to make the absurd assumption that the persons are
spread out over the country like butter on bread. Never-
theless, the statements in their most convenient form are of
great importance for comparing the amount of land and of
air space available in relation to the number of inhabitants,
town by town and country district by country district.

Or, again, we may ascertain that land of certain qualities
can support (say) three persons per acre on the average, and
hence estimate the population that could obtain a living
from a given district.

3. The population of the United Kingdom, June 30, 1907,
is estimated to have been 44,099,000. The number of births
registered for 1907 was 1,147,988 ; the birth-rate was, there-
fore, 26'0 (per thousand of the population per annum). Death-
rates and marriage-rates are calculated in the same way.
The use of these figures is for estimating the future popula-
tion, for observing where the rates are abnormally high or
low, so that, for example, sanitary measures may be taken
with a view to reducing a high death-rate, and for studying
the causes and effects of the fall in the birth- and death-rates
which has been marked in recent years. These rates are
averages of precisely the same nature as the yield of meat in
the first example.

4. If the assessed annual value of the rateable property in
a town is £900,000 and the common expenditure of the town
is £300,000 per annum, a " rate " of 6s. 8d. in the £ would
have to be imposed. Here the expenditure is averaged among
the property-holders in proportion to the value of their pro-
perty. In this case the average (expenditure -=- assessed
value) must be obtained first, and then the sum payable in
respect of each property is calculated.

The national expenditure of the United Kingdom is about
£160,000,000, the population about 45,000,000; the total
national income is estimated as £1,800,000,000, but cannot
at present be known within 10%. On these figures the
necessary tax per head would be £3 lis. if all the money

AVERAGES 17

were collected directly in equal amounts, person by person,
and would be Is. ^d. to 2s. in the £ if it were collected
directly in proportion to income. By such averages an indi-
vidual can estimate whether he is paying his due share of
the national burden.*

The averages so far used are typical examples of arithmetical
averages. An " arithmetical average " is usually defined as
the quotient obtained by dividing the sum of several items
by the number of items; this may be extended to include
the quotient obtained by dividing a total by the number of
persons or things connected with it.

5. If 25 lbs. of tea at 2s. are mixed with 50 lbs. at Is. Qd.,
the cost of the mixture is Is. %d. per lb. Conversely, if the
prices of the constituents and the cost per lb. of the mixture
were given, a simple arithmetic process shows that the
proportions by weight of the constituents were as 1 to 2.

[Weight of dearer : weight of cheaper = Average — price
of cheaper : price of dearer — average.]

If 100 unskilled workmen at 25s. and 50 skilled at 37s. are
employed, the average wage per workman is —

100 X 25s. + 50 X 37s. „„
150 = ^^'-

The last illustration is an example of a "weighted
average," the numbers 100 and 50 being the weights in this
case ; the same process can, of course, be used for combining
several groups.

A " weighted average " is obtained as follows : — Each of a
series of quantities is multiplied by the number of persons or
things connected with it, these multipliers being called
" weights " ; the sum of these products is taken as numerator,
the sum of the weights as denominator ; the fraction is the
weighted average.

Examples and theory f show that slight errors in the

* Actually the problem is very difllcult, since a great part is obtained
in indirect taxation.

t Elements of Statistics, pp. 205-212 and 304.

18 AN ELEMENTARY MANUAL OF STATISTICS

" weights " have little effect on the average, if a fairly large
number of terms are involved, none of them preponderant ;
and it frequently happens that the weights must be estimated,
while the wages (or the other numbers concerned) are known
accurately. Further, it is only necessary to know the ratio
of the weights to each other, as a little consideration will
show. If Wj, Wg' '"'3 ^1"® weights, and %j, n^, n^ numbers,

the weighted average is ^ ^ 11 • ■'■' ^'^^ weights

1 "*" 2 "*" 3

are changed, by multiplying each by h, to kw^ kiv^, hw^, the

. , , , . kwM, + kw„n„ + kWoU. , . , , ,

weighted average is — 7, a. h \ Tc — ^ which clearly

equals the former fraction.

One practical result of this principle is that the weights
may be expressed in round numbers.

Example.

Populations. eSj^rXwTr;. WeeWy Wage.

(1)
16,060 4,123 13s. 6rf.
18,300 4,527 14s. Od.
20,500 4,802 I65. Qd.
22,600 5,432 15s. 6d

Weiglits. Weights.
(2) (S)

4100 8

45 9

48 10

54 11

If weights (1) are taken, the average wage is found to be
14s. 10"0d ; if the round numbers (2) are used, the average
is 14s. \Q'\d. If it is observed that the numbers of labourers
are nearly proportional to the populations, and if the weights
(3), which are also nearly proportional to the populations, are
used, the average is 14s. IQ-^d.

The effect of taking approximate numbers for weights
should always be carefully tested before the result is accepted.

6. In calculating averages of this kind, the work can often
be greatly abbreviated without affecting its accuracy by either
of the methods used in the following example. The proofs
are left to the student.

AVERAGES

Calculation of the Average Wage op the Group whose
Wages are shown in Columns 1 and 2.

4.
Product of

Numbers.

Wages.

Wages 8».

Columus
1 and 3.

Wages 18s.

Product of Columns 1 and 5.

8s.

- 10s.

- 270

10«.

- 8s.

184

lis.

- 7s.

196

12s.

164

- 6s.

246

13s.

225

- 5s.

225

14s.

294

- 4s.

196

15s.

406

- 3s.

174

16s.

488

- 2s.

122

17s.

585

- Is.

18s.

650

19s.

715

+ Is.

— + 65

20s.

780

+ 2s.

— 130

21s.

806

+ 3s.

— 186

22s.

714

+ 4s.

— 204

23s.

720

+ 5s.

— 240

24s.

640

+ 6s.

— 240

25s.

561

+ 7s.

— 231

26s.

378

+ 8s.

— 168

27s.

304

+ 9s.

— 144

30s.*

572

+ 12s.

— 312

889

9132

- 1678 + 1920 = 242

Using Column 4 —

., • Q , 9132
the average wage is 8s. + s.

Using Column 6 —

889

242

18s. S^d.

the average wage is 18s. + ^^^.s. = 18s. S^d.

889

Columns 3 and 5 are equivalent to Column 2. In 3, 8s. is
taken simply because it is the minimum entry. In 5, inspec-
tion of the figures shows that the average is likely to be
between 16s. and 20s. ; 18s. was chosen as the starting point,
as it appeared (without working) to be just below the average;
the nearer the point chosen to the average, the less the
numerical work required.

* Actually, " 28s. or more."

20 AN ELEMENTARY MANUAL OF STATISTICS

The following table is a condensation of the one just given,
and is suitable for rough, but fairly accurate, work.

(")

(b)

Product ot

Wages.

Numliers.

Numbers.

Wages.

In 6s. units.

(a)and(b).

Below 10s.

say 7s.

7s. +0

10s. and below 15s.

186

12s.

15s. „ „ 20s.

314

17s.

20s. „ „ 25s.

266

22s.

25s. „ „ 30s.

27s.

30s.

4-6

204

Average, 7s. + ^^ of 5s. = 18s. 4>d.

Here 12s., I7s., etc., are taken as the middle wages of the
groups 10s. to 14s., 15s. to 19s., etc. If the wages were not
in exact shillings, but were originally given as " 23 persons
earning 10s. and less than lis. " etc., then 12s. 6d., I7s. 6d.,
etc., should be taken for the middle wage of the groups.

7. In distinction to the "arithmetical averages " described in
paragraphs 1-5, which are mainly of use in facilitating further
arithmetical processes, that in paragraph 6 may be called
a descriptive average, for it can be used as an abbreviated
way of describing the " group " of wages in the table.

The following sentences contain nine descriptive * averages.
From the Board of Trade inquiry as to rents, prices and
wages in the towns of the United Kingdom, f we learn that
the average family weekly income was 36s. 10^., the average
number of children Jiving at home was 3-0, the total expendi-
ture on food was 22s. 6d., of which 4s. 5|d and 3s. 7d. were
used for the purchase of 6-5 lbs. of meat and 320 lbs. of
bread and flour respectively. The average rent for a five-
roomed house outside London was about 6s.

* This word is not in general use as a teclinical term, but may be
suggested as useful in classifying averages,
t Cd. 3864 of 1908.

AVERAGES 21

Such averages are usually calculated by adding the total
wages (expenditures, quantities, etc.) and dividing by the
number of instances ; that is, they are arithmetical averages,
or (where the method of paragraph 5 has been used) weighted
averages.

An alternative method of description would be to find out,
e.g. the size of house which was most commonly used by the
working-class ; thus, if we know that 15, 25, 50, and 10 % of the
families inhabited 3-, 4-, 5- and 6-roomed houses respectively,
the 6-roomed house would be most usual or " predominant."
We might further determine that (say) 6s. %d. was the " pre-
dominant " rent. Our whole description might then be
given in terms of "predominant" wages, rents, etc. As a
pure description this is more vivid than the former; we
should be describing the family of which, in fact, there were
most instances, instead of an artificial family with 3"6 children.
Such predominant rates are in statistics regarded as averages,
and are technically called " modes " (fashionable, common).

The " mode " may be defined as that value of the graded
quantity (wages, years, etc.) at which the instances are most
numerous. Very generally in the statistics with which this
book deals the apparent position of the mode depends on
the accident of grading, and the mode cannot be exactly
determined even by mathematical analysis.

Another objection to its general use is that it is not
obtained by a simple arithmetic process, and cannot be
used, like arithmetical averages, for obtaining totals : if the
arithmetical average of 3,000 men's wages is 30s., the total
wage is £4,500, but if we are told only that the "mode"
is 30s. we cannot calculate the total.

The "mode" is more useful in anthropometrical and
biological statistics, where there is a definite type, from
which the measurements of the individuals of a group show
deviations ; in such cases the position of the mode affords
precisely the measurement that defines the type.

Sometimes the word average is restricted to merely arith-
metical measurements, whil§ th$ word mean is used when a

22 AN ELEMENTARY MANUAL OF STATISTICS

group is described ; if this distinction were made, " modes "
and " medians " (see p. 24) would be means. But there is
no general agreement on this point, and French and German
writers do not make a corresponding distinction ; we therefore
regard the words as synonymous.

8. The group of wages given in paragraph 6 is a slightly
modified statement of the weekly wages of women in the
cotton industry. More complete figures are represented on
the adjoining diagram. Such a diagram showing vertically

Women's wages in the Cotton Trade

6000

4000

5 10 15 20

5hilling3 per week

the relative numbers corresponding to the wages (ages, size
or other measurements) marked on tEfe horizontal scale is
known as a "frequency curve,'' and a great part of more
advanced statistics deals with such curves, which show the
frequency of the occurrence of examples at various measure-
ments.* Here we will only observe that the complete
description of a group can only be given by such a curve or
by an elaborate table, and that averages or means are only a
shorthand or abbreviated way of describing some important
characteristics of the group. The arithmetical average, which

'■■ The dotted line in the diagram shows the effect of smoothing off
he angles of the broken line ; the latter represents the data as given.

AVERAGES 23

shows on the horizontal scale the position of the centre of
gravity of the area contained by the curve, and the " mode,"
which shows on the horizontal scale the position of the
highest point, have already been discussed ; in this case we
certainly cannot obtain the latter correct to Id., as we can the
former.

In paragraph 6 we assumed for simplicity that the wages
were exactly at 10s., at lis., etc. More accurately we now
read the column as "10s. and under lis.," "lis. and under
12s.," etc. Women's wages in the cotton trade are to a large
extent piece-rates, calculated out to ^d., and do not tend to
arrive at exact shillings. The arithmetical average is in fact
(as given in the Report, Cd. 4545, p. 28) 18s. 8d., which we
should have obtained if we had assumed that the average
for such a group at "lis. and under 12s." was lis. 4|^d
and so on; actually some of the women are paid exact
shillings, but many are paid by the piece and their earnings
amount to any odd money ; in the illustrative work we took
it as lis., etc. No general rule can be given for such ap-
proximation ; each case must be understood and judged on
its merits.

9. Now make a new table from these figures as follows —

Total (or cumulative) number. Total (or cumulative) number.

Earning under lis.

Earning

under 21s.

592

12s.

22s.

654

13s.

119

23s.

705

14s.

164

24s.

753

15s.

213

25s.

793

16s.

271

26s.

826

17s.

332

27s.

847

18s.

397

28s.

863

19s.

462

All

889

20s.

527

Consider the values of a, b, c in the following statements :
" Half the wage-earners received a/- or less, one quarter received
hj- or less, one quarter received cj- or more."

To determine a we want the position of the 445th worker
(in order of wages from the beginning). The 397th worker
just failed to reach I8s, ; but 65 earned from 18s. to 19s,

24 AN ELEMENTARY MANUAL OF STATISTICS

and we need the 48th up this group. Making the not un-
reasonable assumption * that 65 were distributed uniformly
Id. by Id from 18s. to 19s., we find that the 48th was at
18s. 9d.

The work may be shown as follows —

«/- = 18s. + '*^^ 7 ^^^ of Is. = IBs. 9d.,
' 6o

similarly, Ij- = 15s. + ^^^ ~ ^^^ of Is. = 15s. 2d.,
similarly, cj- = 22s. + t^^ ~ ^^ of Is. = 22s. M.

a/- is called the " median," i/- and c/- are the lower and
upper " quartiles " for this wage group.

The median and quartiles of a group may be thus defined :
If the members of the group are ranked in order according
to the measurement (wages, ages, height, etc.) under con-
sideration, then the measurements of the members most
nearly one quarter, one half and three quarters respectively
along the rank are the " lower quartile," the " median " and
the " upper quartile."

Such quantities obviously afford a very simple and definite
description of a group. In fact, this method is the most
helpful of the statistical abbreviations, and it is rapidly
coming into common use.

The main objection to the median, as to the "mode," is
that it does not lend itself to further numerical work. The
following statement is true of the arithmetical average, but
not necessarily of the median or mode —

If ftj, a^ are the average wages of two groups of n^, n^

persons, then -LJ ^-? is the average for the combined

group.

10. Fallacies — i. The average rate of a journey where altern-
ate miles are done at 8 and 1 2 miles per hour, is not 10 miles

* Actually there is some concentration at 18s. ; with full information
this should be taken into account.

AVERAGES 25

per hour, but 9'6 miles per hour, for two successive miles
occupy 12^ minutes.* The average rate of increase when
three successive annual increments are 20 V , 30 °/ and 40 °/ ,

/O' /o /o'

is not 30 %. but

4/(1-20 X 1-30 X 1-40) X 100 1 - 100 = 29-75 %.

ii. The average rate of interest of three sums of money
bearing 3, 4 and 5 ^ respectively, is not necessarily
-1(3 + 4 + 5) = 4 % ; e. ^r. if the sums are £1,000, £3,000 and
£8,000 respectively, the interests are £30, £120 and £400,
and the average rate is '^^■^ ^. " Weights " cannot be
neglected without examination, nor unless certain special
conditions are satisfied.

iii. If three groups of men have their wages raised each
20 y^, the average is not necessarily also raised 20 % unless the
relative numbers in the groups are unchanged. This is shown
by the following example in which the average actually
falls—

A 1^ n'TiTom

KATE.

At second Date (Wages increased

Ai FlKal

20%, but relative Nos.

changed).

Numbers.

Wages.

Numbers.

Wages

Group 1.

100

20s.

400

24.S.

Group 2.

200

25s.

200

30s.

Group 3.

400

40s.

100

48s.

Total 700 Average 34fs. Total 700 Average 29fs.

Neglect of a change of weights always distorts and some-
times reverses the results.

Again—

POPDLATION A.

Number. Death-rate.

Population B.
Number. Death-rate

Components :
Under 5 years
Over 5 years .

44,000

4,000
40,000

16-4

25-5
15-5

44,000

1,000
43,000

16-2

26-0
16-0

Here the death-rate of Population A as a whole is higher
than that of B, though the rates of the two parts shown are
each lower ; for A contains a larger proportion of young
children, for whom the rate is high. J

* 9-6 is the harmonic mean between 8 and 12.
■j- The geometric mean.

X In connection with this example see the method of correcting the
death-rate, p. 106, below.

26 AN ELEMENTARY MANUAL OF STATISTICS

In using arithmetic averages for the comparison of two
groups, it is necessary to analyse the groups, and find if they
are sufficiently homogeneous (of the same kind) in themselves,
to allow a reasonable comparison.

iv. False accuracy. — The average wage of two groups, the
first of 100 men whose average is stated to be between 25s.
and 26s., the second of 200 men whose wage is between 30s.
and 31s., is not known to be

100 X 25/6 + 200 X 30/6 „„ ^ „ ,
300 ^^'- ^"''■'

but is only known as between 28s. M. and 29s. M. Where
there are many items, the average is more accurate than its
constituents, but not necessarily when there are only two or
three.

11. In an average the constituents of the numerator
should be similar in kind to each other, and so should the
constituents of the denominator. Also the various parts of
the denominator should bear similar relations to the parts of
the numerator. It is thus correct to speak of the death-rate
of a population of healthy male adults, for they are subject to
similar risks ; it is correct to speak of the average wage of
men in a trade. As we extend our view to include the
whole population or a large group of trades, more and more
caution is needed in the use of the average, though there are
problems in which these wide averages are useful. It is
doubtful whether any use can be made of the average fre-
quently stated : " Total imports and exports divided by the
population," as measuring the amount of foreign trade ; for
imports and exports are of different, even opposite, kinds for
most practical purposes, and do not concern equally all the
members of a population. Similarly "the average income
per head of the population" can only be used for arithmetical
purposes, not (except in a few cases) for comparison of gn^
population with another.

CHAPTER IV

THE ACCUEACY OF AVERAGING AND OTHER ARITHMETICAL
PROCESSES

Population op

THE County op London

(1)

(2)

(3)

(4)

(5)

(6)

Enumerated

Nearest 1000

Next 1000 under

1851

1901

1851

1901

1851

1901

OOO's

City of London .

127,869

26,923

128

127

Battersea . .

10,560

168,907

169

168

Bermondsey

85,308

130,760

131

130

Bethnal Green

90,193

129,680

130

129

Oamberwell

54,667

259,339

259

Chelsea . . .

54,078

73,842

Deptford . .

24,899

110,398

110

Finsbury . .

125,418

101,463

125

101

125

101

Fulham . . .

11,886

137,289

137

Greenwich . .

47,377

95,770

Hackney . .

53,589

219,272

219

Hammersmith .

17,760

112,239

112

Hampstead . .

11,986

81,942

Holbom . . .

95,676

59,405

Islington . .

95,329

334,991

335

334

Kensington . .

44,403

176,628

177

176

Lambeth . . .

139,325

301,895

139

302

139

301

Lewisham . .

18,616

127,495

127

Paddington . .

48,415

143,976

144

143

Poplar . . .

47,162

168,822

169

168

St. Marylebone

157,696

133,301

158

133

157

133

St. Pancras . .

166,956

235,317

167

235

166

235

Shoreditch . .

109,257

118,637

109

119

109

118

Southwark . .

. 152,371

206,180

152

206

152

206

Stepney . . .

238,910

298,600

239

299

238

298

Stoke Newington

6,076

51,247

Wandsworth .

40,204

232,034

232

Westminster .

. 244,178

183,011

244

183

244

183

Woolwich . .

43,177

117,178

117

Total

of the 29 districts

2,363,341

4,536,541

2,362

4,535

2,349

4,521

Averages. . .

81,494

156,432

81,45"

156,38"

81,00" 155,9""

Ratios 1851 to 189]

1000 :

1920

1000

1920

1000 :

1925

28 AN ELEMENTARY MANUAL OF STATISTICS

1. The word error is used in statistics, not as meaning a
mistake, but as denoting the difference between an estimate
and the generally unknown exact measurement. We must
distinguish between two methods of measuring error. In the
adjoining table, the population of the county of London is
shown as 4,.536,041 in column (2), and estimated as 4,535,000
ia colunm (3) ; the difference, 1,541, is called the absolute
error ; the ratio of 1,541 to 4,535,000, i. e. -00034, is called
the relative error. The relative error may also be expressed
as a percentage error, in this case '034 ^. No simple rule
can be assigned as to when absolute and when relative errors
are the more important.

In the table, columns (1) and (2) give the populations of
the city of London and the Metropolitan Boroughs in 1851
and 1901. Columns (3) and (4) give the same numbers to
the nearest thousand; columns (5) and (6) give the same
numbers omitting the last three figures in each case.

Example of absolute errors. — The average of the successive
numbers to 999 is 499-5. In numbers stated as in columns

(5) and (6) we are equally likely to have omitted any number
from to 999, and are liable to an absolute error which can-
not be greater than 29 X 999 or less than 0, and whose most
probable value is 29 x 4995 = 14,500 (nearly). The errors
are actually 13,341 and 15,541, as may be seen from columns
(1) and (2).

Example of relative errors. — The relative errors in column

(6) are the ratio of numbers varying from to 999, with
average value very nearly 500, to the numbers in the
column (26,000, 168,000, etc.). The smaller the population
the greater the probable relative error. In the first line it is
nearly ^\, while for Stepney it cannot be so great as ^g.
There is no simple relation between the relative errors in the
items and in the total, except that the latter is between the
greatest and least of the former.

2. It is clear that columns (5) and (6) under-estimate all
the items and the total, while columns (3) and (4) are equally
likely to be in excess and defect, Such errors as the latter

THE ACCURACY OF AVERAGING 29

are called fortuitous or unbiassed errors, while the former
(which all tend in the same direction) are Massed. The
simple total of the absolute biassed errors in column (6) is
the absolute error in the total. The case is very different
for the unbiassed errors of columns (3) and (4). It is just as
likely that they will be subtractive as additive ; actually 14
of the numbers in column (4) and 13 in column (3) are in
excess, while 15 and 16 respectively are in defect. It is
obvious that these errors possess a strong tendency to neutral-
ize each other, but it is not obvious to what extent this
neutralization will take place.

[Paragraphs 3 and 5 can be omitted without losing the
sequence of the other paragraphs.]

3. The following rules must be accepted at present with-
out proof, but they certainly appear plausible, and can be
confirmed by experiment.

In the case of unbiassed errors —

''a) The absolute error in the total increases with the number
of items, when each is subject to the same unbiassed
absolute error.
(&) The best estimate for the absolute error in the total is
the average absolute error to which the items are
liable, multiplied by the square root of the number
of items.

number of items.

(d) The best estimate for this relative error is the average

absolute error of the items multiplied by the square
root of the number of items and divided by the
total.

(e) It is better to write (d) : — The best estimate for the

relative error of the total is the average absolute
error of the items divided by the average of the
items, and also by the square root of the number of
items.
Examples of (a) and (c). — If the first 4 lines only of
column (i) are added, the absolute and relative errors are

30 AN ELEMENTARY MANUAL OF STATISTICS

respectively 730 and ■^, while those for the 29 lines are
1,541 and -^Vr-

Hxamples of (b). — The average absolute error to which the
items in col. 4 are liable is very nearly 250, all numbers from
to 500 being equally probable in the table above. The best
estimate for the error of the sum (if we know nothing further
about it) is 260 ^^29 = 1,346, and the sum may be written
4,535,000 ± 1,346.* Actually column (2) shows that the true
value is just outside this margin. The total for column
(1) is just inside the similar margin (2,362,000 ± 1,346)
obtained from column (3). We must not expect in general
to be just at the margin.

Examples of (d) and (e). — The average of the items in

column (4) is 156,400 ; their average absolute error is 250 ;

their number 29. The relative error in the total is then

estimated

. , ,, 250^29 , . , , 250 .

from (d) as ^^^^^i^, and from (e) as J^^^^^^^, ^mce

156,400 is the average item. Each of these = -0003. The
relative error found by comparing columns (2) and (4) is '00034.

Similarly the computed relative error in column (3) is
■0006, and that found from columns (1) and (3) is '0005.

Of course there is no means of determining what the error
actually is when we only know the estimates. These rules only
afford a means of estimating the errors to which we are liable.

4. The averages given in the last line but one of the table
are of no importance except for illustrating the principles of
this chapter.

It is evident that the absolute error of the average equals
the absolute error of the total divided by the number of items,
in this case 29.

It should also be evident that the relative error of
the average is exactly equal to the relative error of the total.

* More exactly this means, "it is as likely as not that the total is
within these limits, and very unlikely that it is as much as (say) six
times as far from the estimate (4,535,000) as these limits are. The most
probable value ia 4,535,000, in the absence of information."

THE ACCURACY OF AVERAGING 31

Biassed errors then remain in the average unaltered. The
absolute error of the average will be very near the average
absolute error of the items. Thus for both columns (5) and
(6) the average errors may be expected to be (see paragraph
1) 500. We should therefore estimate the averages as
81,000 + 500 = 81,500 for 1851, and 155,900 + 500 =
156,400 for 1901, and these estimates differ very little from
those shown in columns (1) and (2).

Unbiassed errors tend to disappear in the average just as
they tend to disappear in the total. In fact, the absolute
errors in the averages of columns (3) and (4) are only 44 and
52, and the relative errors '0005 and -00034 respectively.

5. The rules of paragraph 3 become for averages —
In the case of unbiassed errors —

(h) The best estimate for the absolute error of an average
is the average absolute error of the items divided by the

250
square root of their number ; viz. — .= = 46.

x/29

(e) The best estimate for the relative error of an average

is the average absolute error of the items divided by the

average of the items and also by the square root of their

250
number, viz. =-— r— ttt^ — t=^ = '0003 as before.
156,400^29

6. As a further illustration of biassed errors it may be
noted that to obtain round numbers in a long addition of
n items, we may carry •45w from the unit column to the tens,
instead of doing the addition, since 4'5 is the average of the
digits to 9. From the hundreds column we may carry •5w,
since 50 is very nearly the average of the numbers to 99.
Similarly in adding money we may add 5|d x n for the
pence, and 9s. 6d. x n for the shillings, if the items end in
pence and shillings respectively. If both pence and shillings
are given we add 10s. x w to the £.

Thus in column (1) by this rule the numbers to carry
would be 4-5 x 29 = 13, -5 X 29 = 14 or 15. Actually the
numbers carried are 16, 16, 14, 15 in order.

32 AN ELEMENTARY MANUAL OF STATISTICS

7. Comparison of similar totals or averages. — Here we
only deal ■with relative error. The actual ratio of growth
shown in columns (1) and (2) is 1 : 1-920. That shown in
columns (5) and (6) is 1 : 1-925. The relative error is

Yq^ = '0026. The relative error is identical for averages

and for totals.

Tlie relative error of the ratio is very nearly equal to the
difference between the relative errors of the two terms. If
the errors are both positive or both negative, as is the case
with biassed errors (unless there is a change of bias), the
error in the ratio is less than that of the terms. Thus the
relative errors for the totals of columns (5) and (6) are '0061
and "0032 respectively, both in defect ; the difference is '0029
very nearly the same as "0026 just given.

There is no reason to expect that the small errors resulting
from the addition of unbiassed errors will be both in excess
or both in defect, though it happens to be the case in
columns (3) and (4). In general we may expect the error
resulting from unbiassed errors to be slightly greater in the
ratio than in the terms.

The general result is that unbiassed errors tend to disap-
pear in the averages and not to reappear in the ratio, while
biassed errors tend to disappear in the ratio. The comparison
of averages well constructed on similar principles generally
has great accuracy, greater than that of the original items or
totals. It has already been pointed out that the process of
" weighting " also leads to accuracy. In fact, the ratio of
weighted averages can under certain conditions which are
often realized be obtained with a surprising accuracy. It
can generally be determined by experimenting with the
numbers whether these conditions are present.*

8. In dealing with a group, as in the last chapter, it is to
be noticed that there may be a good deal of uncertainty
about the extreme parts of the group, and yet the averages
may be well determined. Thus the " mode " is not influenced

* For an example, see Statistical Journal, 1906, pp. 164 seq.

THE ACCURACY OF AVERAGING 33

at all by anything except the central portion. The median
is known completely for the table on p. 19, if the numbers
(say) above 25s. and below 15s. are given, but not the
exact wages in these marginal groups, and if numbers and
wages are given in the central region ; even if the top
group, 26 at 28s. or more, were dropped out entirely, the
median would only be lowered from 18s. 9d. to 18s. 7c?.
The arithmetical average is more easily affected by the
position and magnitude of the extremes, especially the
upper extreme ; if of the 26 at 28s. or more (whose average,
in fact, is near 30s.), 6 were at 35s., and 10 each at 40s. and
at 50s., the average would be raised from 18s. 8d. to 19s. Id. ;
in such cases, general knowledge of the structure of the
group will often make possible the assignment of narrow
limits within which the average must lie.

9. When only two or three or a few terms are present, the
rules given as to approximate work and round numbers in
Chapter II apply. The greatest absolute error in the terms
of an addition or subtraction, or in a factor of a product,
dominates the error in the result. Many terms (say 20 or
more) are necessary before the fortuitous errors can be confi-
dently expected to neutralize each other. Of course, paragraph
7 above applies if the two terms form a ratio. The general
practical rule in all cases involving few terms is to work
through the problem, assuming every error is as great as
possible under the conditions of the question, the sign of the
errors being so chosen that they all work towards increasing
the error in the result. Then give the answer in one of the
forms of p. 7 ; if suflficient accuracy for practical purposes can
be attained by giving the nearest round number which is
certain, the statement "correct to the last digit given" is
the best.

10. That a small absolute error in an item may have a
great effect on the result may be illustrated by the following
examples —

34 AN ELEMENTARY MANUAL OF STATISTICS

(a) Cost of workmen's budget.

Prices.
1st date. 2nd date.

Meat, 81bs. . . . 8M 9^^.

Bread, 201bs. . . . 2|rf. 2|d

Total .... 9s. IQd. 10s. Id

Suppose that the price of bread had been obtained as an
average 2'S8d., and had then been written to the nearest
farthing, viz. as 2^d.

Now suppose that a slight mistake had been made in the
working of the average for this price, everything else being
correct, and in fact it should have been 2"37d Given to the
nearest farthing this is 2Jc?. The first budget would then
have amounted to 9s. 5d., and the increase would have ap-
peared as + 8 %. In this case a relative error of not more
than 1 in 200 results in a relative error of 5 in 3.

A careful writer would have said in this case that there
was no certainty of any change in the total.

(b) Of 695,720 members of Trade Unions, 7-4% were
unemployed at the end of September 1909. Seven groups
of trades account for 579,899 members, of whom 8'5 ^ were
unemployed. Can the number be deduced for the remaining
group ?

At first sight we might proceed as follows —

Members. Unemployed.

7-4 % of 695,720 = 51,483
8'5% „ 579,899 = 49,291

Eesidue 115,821 2,192

But the 7'4 and 8-5 are more exactly from the original
figures 7'42 and 8-455, the total number unemployed was
51,749, and that for the seven groups 49,028. The residual
number was therefore 2,721, which exceeds the estimate
(2,192) by 25%.

CHAPTER V

USE OF DIAGRAMS

1. Diagrams do not add anything to the meaning of
statistics, but when drawn and studied intelligently they
bring to view the salient characteristics of groups and series ;
they show the various parts in relation to each other and
to the whole, bring to light the unity that underlies the
scattered figures, and suggest in what directions investiga-
tion is needed. Merely pictorial diagrams are not only
unlikely to be of much use, but in advertisements and
political propaganda are often deliberately misleading, though
literally correct. In the author's opinion the graphic method
should rarely be used except (i) to show the relations of one
part of a group to another (the word used in the sense of
p. 1, where the various members differ in respect of one
measurable characteristic), (ii) to exhibit a series of similar
estimates date by date, (iii) to compare two or more groups,
(iv) to compare two or more series, (v) to exhibit three
relations which can be geometrically united.

Diagrams which simply show relative magnitudes — e. g. the
populations of three countries at one date, or two isolated
figures, such as the sale of some commodity at two dates, where
the horizontal scale shows no graduated quantity (time, age,
wage, height, etc.) — are of no assistance for the comprehension
of the numbers.

Nevertheless, a skilful writer can often devise statistical
diagrams of other kinds which help the visualization of a
complex argument, and the aid received from diagrams varies
greatly from person to person, so that it would be rash to lay
down too rigid rules.

D 2 35

36 AN ELEMENTARY MANUAL OF STATISTICS

2. The four pages of diagrams given illustrate the main
principles of graphic statistics and afford examples of most
of the methods that are to be recommended. The first shows
the two ways of representing a group; one is chosen that
presents some difficulties, in order to show the elasticity of
the method.

Ages of Men and Boys Employed in Coal Mines,
England and Wales, 1901.

Age.

Number.

Per Mille.

Cnmulative

10-14 years

2,761

Under 14 years 7

14-15

3,992

15-20

36,469

106

20-25

67,349

164

270

25-35

131,818

322

592

35-45

86,735

212

804

45-55

53,305

130

934

55-65

22,073

988

65-75

,4,645

999

75 and over

382

1
= 1,000

Total

1,000

409,529

The ordinary way of showing such a group graphically is
that of B on the page opposite. The years are marked off
on a horizontal scale. The numbers in the six equal age
periods (15 — 25 years, 25 — 35 years, etc.) are represented by
rectangles proportional to these numbers on any convenient
vertical scale. It is customary, but inaccurate, to join the
middle points {a, h, c, d, e, f) on the tops of these rectangles
by straight lines, as in the figure. If there are many narrow
rectangles, as in the diagram (p. 22) above, the inaccuracy is
slight, and may be ignored.

In diagram B j^jj-th inch square represents 4 per 1,000 of
the persons throughout. It is not difficult to see that if we
represent the numbers at 15 — 20 years and 20 — 25 years
separately, we must keep the same areal relation by doubling
the vertical scale. Similarly, if the number at 14 — 15 years
were shown, it would be represented by a vertical scale
increased tenfold. This method will become clear as soon as
an attempt is made to draw the diagram from the numbers.

11 Ages of Coal-miners.

A Cumulative diagram showing tlie fotal number whose age is
under each age marked on the horizonfal scale:—
e.g. 804 (per 1000) shown as Phi, are under 45 years.

lOOO
800

600

M
400

200

10 15 20 25 30 35 40 45 50 55 60 65 70 YcarS

i ' ' Py^

-""^^ i '

! ! Y'

' .' c/ :'

i /;

Diagram showing distribution by Age.

15 20 25 30 35 40 45 50 55 60 65 70 Years

The rectangular blocks show the numbers in lOyeargroups.
The dotted rectangles show the five-year groups from lOto 25 years.
The curve shows the number at each yean

38 AN ELEMENTARY MANUAL OF STATISTICS

Since there can be no sharp division of numbers as we pass
from age to age, the apparent division being introduced by
the accidental placing of the age limits, it is clear that the
whole group should be represented by an unbroken line.
The ordinary introduction of such a line as ahcdef is in-
tended for this purpose. A little reflection will, however,
show that we should keep the area standing on any given
base unchanged, and that this line cuts off a great part of the
area on 25 — 35 years. To avoid this a freehand curve should
be carefully drawn, so as to keep all the areas unchanged. It
will at once be seen that in this case the information is not
sufficient for an accurate drawing, and that there is something
arbitrary in the figure. If it is found that the line is not
definitely placed, the figure should be left in the original
rectangles.

Finally, the extremities of the curve, below 14 and above
70 years, must be drawn to satisfy the conditions of the data
(in this case there are no children under 12 or 13 years), the
continuity of the drawing being preserved.

The vertical scale adds very little to the information, and
might in this case be removed after the drawing is complete
with little loss.

These difficulties are present to some extent in all group
diagrams.

3. Diagram A represents the cumulative numbers in the
last column of the table, p. 86. The dots (K, L, P, etc.) show
the information exactly as it is given, and there is no element
of approximation or arbitrariness.

In the figure these dots are joined by straight lines. To
obtain a more perfect representation the angles at K, L, etc.,
should be rounded off by a careful freehand curve, for there
is no reason why the line should be broken exactly at 20, 25,
30 years, etc. The number up to any assigned age may then
be read from the freehand curve. [To avoid confusion it is
not drawn in the figure.]

It will be found that the curve cannot be finished at either
end without further information.

USE OF DIAGRAMS 39

The quartiles and median (see p. 24) of the group may
readily be found approximately from the drawing. The line
MC is drawn horizontally through the middle point of the
vertical scale to meet the freehand curve at C ; CD is then
drawn vertically to meet the horizontal scale at D. The
reading at D (32 years) is the median.

The mode is the reading above which such a curve is
steepest, but is not easily determined with accuracy by the
eye, and needs mathematical analysis before an exact value
can be obtained.

A diagram of this kind is more accurate and useful than
such as B, and is more easily used for the comparison of two
groups. It requires practice to grasp its meaning readily.

4 The final test of the goodness of a diagram is its
legibility and clearness of meaning.* The diagram should
carry on its face a suflScient definition of the facts represented.
There should never be many lines in one diagram, unless they
can be kept apart from each other. Lines should be dis-
tinguished by colours or clear hachetting, and, where suitable,
their meaning (e.g. "weight" and "value," p. 44) should be
written close to them. Cross references should be avoided.
If there is much detail, either the data should be separated
into two or more diagrams, or the numbers should be left in
a table and not represented graphically. An overloaded
diagram defeats the only purpose for which it is intended.

Any diagram can be drawn on the back of a postage stamp
or enlarged to cover a wall. The page of a book is generally
sufficient for all the detail that ought to be shown, and large
sheets and folded pages are to be avoided. The ratio of the
vertical to the horizontal scale must be chosen so as to bring
out those fluctuations or movements which are the subject of
study; then the absolute scale should be so chosen that the
allotted space is occupied.

5. The following diagrams show the method of representing

* Diagram B above is, of course, only intended to show the method
of working. The other diagrams printed in this book satisfy, it is
hoped, the conditions here laid down.

40 AN ELEMENTARY MANUAL OF STATISTICS

Average Annual

Gazette Price of

Qainquennial

DlffproTir-BS

Wheat per

Averages.

Quarter.

1864

40'2s.

1865

41 •8s.

—

1866

49 -98.

1864-1868

52-08.

—

2-ls.

1867

64-4S.

1865-1869

53-6s.

+ 10-8s.

1868

63-7s.

1866-1870

bi-GK.

9-l8.

1869

48 -28.

1867-1871

56-Os.

7-8s.

1870

46-8S.

1868-1872

54-58.

7-7s.

1871

56-7s.

1869-1873

53-5S.

3-2s.

1872

57 -Os.

1870-1874

55-Os.

2-08.

1873

58 -78.

1871-1875

54-7s.

4-08.

1874

55 -78.

1872-1876

52 -6s.

3-ls.

1875

45-2S.

1873-1877

52-58.

7-3s.

1876

46 -28.

1874-1878

50 -Os.

3-8s.

1877

56-7s.

1875-1879

47-7s.

9-Os.

1878

46-4s.

1876-1880

47-58.

1-ls.

1879

43-8s.

1877-1881

47-3S.

3-5s.

1880

44-38.

1878-1882

45-Os.

0-78.

1881

45-3s.

1879-1883

44-Os.

l-3s.

1882

45-ls.

1880-1884

42-48.

2-78.

1883

41-68.

1881-1885

40-18.

1-58.

1884

35 -7s.

1882-1886

37 -2s.

l-5s.

1885

32-8s.

1883-1887

34-78.

1-98.

1886

31-Os.

1884-1888

32-8s.

—

l-8s.

1887

32-5s.

1885-1889

31 -6s.

0-9s.

1888

31 -Ss.

1886-1890

31 -48.

0-48.

1889

29-7s.

1887-1891

32-68.

2 -98.

1890

31 -98.

1888-1892

32-18.

0-28.

1891

37-08.

1889-1893

31-Os.

6-Os.

1892

30-2S.

1890-1894

29-6s.

0-6s.

1893

26 -3s.

1891-1895

27 -9s.

l-6s.

1894

22 -88.

1892-1896

25-78.

2-98.

1895

23-ls.

1893-1897

25 -7s.

2 -6s.

1896

26-2s.

1894-1898

27-28.

1-Os.

1897

30-2s.

1895-1899

27-8s.

2-4s.

1898

34-Os.

1896-1900

28-68.

5-4s.

1899

25-78.

1897-1901

28-7s.

3-08.

1900

26-9s.

1898-1902

28 -3s.

1-48.

1901

26-7s.

1899-1903

26 -8s.

O-ls.

1902

28-ls.

1900-1904

27-3S.

0-8s.

1903

26-78.

1901-1905

27-98.

1-28.

1904

28-38.

1902-1906

28-28.

O-ls.

1905

29-7s.

1903-1907

28-75.

1-Os.

1906

28-2s.

—

1907

30-6s.

—

USE OP DIAGRAMS 41

a series. In a series we have generally to study both the
short-period fluctuations (regular or irregular) and the
general movement or tendency, or "trend," as it may be
called. In Diagram A (p. 42) the jagged line shows the
data as given. It is at once clear that we have a succession
of rapid fluctuations combined with a general movement
mainly downwards. The problem is to disentangle the
" trend " from the fluctuations. The table on p. 40 shows
how it may be done.

A period is selected, long enough to remove the fluctua-
tions of separate years, short enough to allow a long series of
averages to be obtained.* As in the second column of the
table, averages are taken again and again, and the line of
" moving average " is shown in the diagram. The angles
and small fluctuations of this line should then be smoothed
away, as they are accidental. This smoothed line shows the
trend; in this case it is downward from about 1870 to about
1895, and nearly neutral with some inclination to rise in the
most recent years. This line cannot begin at the beginning,
or end at the end, of the period covered by the data, for
several years are necessary to establish " the trend."

It is now assumed that the smoothed line represents the
course of the events, as determined by slow-acting, cumula-
tive influences, and that the deviations from it are due to
short-period (or, in some cases, accidental) causes. The
deviations, or differences between the price of a particular
year and the average price of the five years of which that
year is the middle, are obtained in the table and represented
in Diagram B ; a new vertical scale is taken to throw the
fluctuations into relief

The smoothed line of Diagram A and the line of Diagram B
show the "trend" and the "fluctuations"; but it is advisable
to preserve also the jagged line of the first diagram.

* If the fluctuations occupy the same length of time (e. g. 10 j'ears),
again and again, this period (10 years) should be taken for the succes-
sive averages. It is not necessary to use the same period throughout
the series.

A Price of wheaf year by year ,

Quinquennial averages — & smoothed line-

Differences from moving average

USE OF DIAGRAMS 43

There is something arbitrary in Diagram B, since the mag-
nitudes of the differences, and sometimes even their sign,
depend on the length of the period taken for averaging.

6. Series can be distinguished by the nature of their
" trends " and fluctuations ; and it is extremely important
to know both these with regard to any series used. The
trends may be up or down, rapid or slow, uniform or chang-
ing. The fluctuations may be periodic (regular) or random
(irregular), great or small. When we have examined the
series, with the help of a diagram, over many years, we may
know what to expect from the phenomena considered ; we
shall be able to tell whether a tendency observed is of a
permanent character, and to distinguish between fluctuations
which are natural to the series and those which show some
great and new disturbance. For example, from this series
we notice that the price fluctuates greatly from causes
which may be present at any time, and that it would be
quite impossible to trace the efl'ect of (say) the introduction
of some small new area into the world's supply, or the effect
of a shilling import duty.

7. In order to show the relations of two terms of a series,
or the size of the fluctuations relative to the total amount,
it is essential to have a visible horizontal line through the
zero of the vertical scale ; otherwise continual and confusing
reference must be made to the numbers on the vertical scale.
This can be realized if the zero line of Diagram A opposite is
covered up, and we ask if the price was halved between 1870
and 1890, or whether the fall from 1891 to 1894 diminished
the price by one- third.

8. The next series of diagrams is designed to illustrate the
danger of ignoring the zero line, some of the fallacies which
an unscrupulous use of diagrams may render plausible, and
the general method of comparing graphically two series.
Figure E, on p. 44, shows the value and weight of iron
and steel exports year by year on a scale which would natur-
ally be adopted. There is no essential reason, however,
why £10 and 5 tons should be represented by the same

N Exports of Iron & 5feel Manufactures

Weight — Value

USE Of DIAGRAMS

vertical distance. In the three figures A, B, C, the weight
is represented on a uniform scale, viz. half that of E ; but in
A the scale for value is so chosen that the lines begin
together, and also (as it happens) the averages for the eleven
years of value and of weight are represented at very nearly

Exports of Ieon and Steel and Manufactures Thereof,
Produce of United Kingdom

Relative Numbers.

Relative Numbers.!

Years. Value

Weight.
Tons.

Value

Weight.

Value

Weight.

Value.

Weight.

OOOO's

OOO'S."

1893 20,26

2,738

52+

81+

1894 18,47

2,566

- 5

1895 19,43

2,738

- 3

1896 23,46

3,423

102

+ 8

+ 21

1897 24,41

3,599

107

+ 10

+ 26

1898 22,39

3,160

+ 5

+ 13

1899 27,71

3,601

111

107

+ 19

+ 24

1900 31,62

3,447

126

,103

111

103

+ 29

+ 22

1901 25,01

2,813

100

+ 13

+ 3

1902 28,88

3,474

116

104

102

104

+ 23

1903 30,40

3,565

122
99

106
95

106

+ 25

Average £24,73

3,193

the same height. In B the equation is made for the last
year, 1903.* Both these are correct, but method B very
frequently gives the better perspective for two series. In
long series it is best not to equate individual years, but to
equate the averages of the last few years given.

C and D are incorrect; the lines for value and weight
are accurate separately, but the zeros of the vertical scales
are not in the same position. It is a simple' arithmetical

* To obtain the working figures for A, take 81 (a number convenient
for numerical work in this case) to represent the value in 1893, and
obtain proportionate numbers for the other years with a slide rule or
otherwise. Take the same number to represent the weight in 1893,
and finish the column by proportion. In this case easy arithmetic
is obtained by multiplying the value by 4 and the weight by 3 less
about 1 %. For B the same weight numbers are used, but the value
in 1903 is equated to 106.

46 AN ELEMENTARY MANUAL OF STATISTICS

problem, of which part of the working is given in the table
above,* to force the lines to begin and end together. D is
merely C enlarged vertically.

A comparison of these five diagrams shows that almost
any appearance may be given to fluctuations by a deliberate
choice of scales, and suggests the need of care and intelligence
in reading diagrams.

9. The following diagram shows one of the few methods
of pictorial work that can be recommended. The proportion
of the parts of a group to each other and the whole are
shown by the sectors of a circle ; since the areas of sectors
are proportional to their angles at the centre and the arcs
on which they stand, there is no possibility of confusing
linear and areal proportions. For the comparison of two
groups, two circles are constructed so that their areas are
in the ratio of the numbers in the groups. It is at once
clear by comparing the angles that the proportion {e.g.) of
males between 14 and 15 years is smaller than that of

NtTMBEE AND AgBS OP PERSONS OCCUPIED IN THE TEXTILE TeADES

OF England and Wales (Including Dealers), 1901

Ages.

Males.

Females.

10-14
14-15
15-20
20-45
45 and over

Number.
24,700
18,332
81,200
267,168
100,775

Eelative No.
18"
13
59
196
74

360"

Number.

30,367

31,402

188,125

359,976

53,352

Belative No.

16«

17
102
196

492,175

663,222

360"

Trt^ = 4'92175 ; hence r = 1-252 inches (1 sq. in. represents 100,000
persons),

TTT^ = 6-63222 ; hence r = l'453 inches (1 sq. in. represents 100,000
persons),
where r stands for the radius of the circle in each case.

* The first and last figures for weight in A differ by 25. Equate the
difference between the first and last figures for value (viz. 122 — 81 = 41)
to 25, and reduce all the value numlaers to the ratio 41 : 25 ; the first
becomes 52, the last 77. Hence the numbers in the table.

Number & ages of Textile operatives

Y" tS 10 20 Ye ar/)

Females

to 15 Years

Males
square inch represents loo.ooo persons.

48 AN ELEMENTARY MANUAL OF STATISTICS

females ; and observation of the areas suggests, (e. ^.) that the
number of women 20 — 45 years is about equal to that of all
men over 20 years.

10. The commoner mistakes made in the construction and
use of diagrams are as follow —

(a) By an injudicious choice of vertical scale the fluctua-
tions are exaggerated (D, page 44), or, on the other hand,
made inconspicuous (E, page 44, value line).

(&) An exaggerated vertical scale has the effect of making
too conspicuous a single year in which the rise was greatest.
It may easily happen that with monthly figures the high
values would be seen to be spread over both the adjacent
years.

(c) When two series are represented on one diagram the
equation is made between an exceptionally high year in
one and an exceptionally low year in the other, with the
result that the relative growths are distorted.

{d) It is not always realized that in such diagrams as B
(p. 42) and E (p. 44), the dot representing the number is to
be placed over the centre of the horizontal distance showing
the corresponding period ; while in A (p. 37) the dot is at
the end of the period. Similarly the dots showing a moving
average (A, p. 42) should be exactly at the centres of the
periods for which the average is taken.

(e) In pictorial diagrams (such as the " big and little loaf") it
is seldom clear whether the linear, areal, or cubic dimensions
are intended to be compared. If one quantity is IJ times
another, for linear comparison the ratios should be 1'5 : 1, for
areal 1'225 : 1, and for volume 1-145 : 1. The three diagrams
opposite illustrate the same ratio 2 : 3 in three ways

USE OF DIAGRAMS

Ratio

2 :3

Lines

Areas

Vo I u m e s

CHAPTER VI

TABULATION

1. Tabulation is the intermediate process between the
accumulation of data, in whatever form they are obtained,
and the final reasoned account of the results shown by the
statistics. The process of tabulation is essentially the selec-
tion from the data of all the persons or things, which have
certain defined characteristics A, B, C, D, etc., and their sub-
division according to other variable characteristics E^, E^, Eg,
etc., and Fj, Fg, Fg, etc. Then ABCD (e. g. Cotton industry,
weaving, men, 4 looTns, in the table below) is the heading of
the table; Ej, Eg, etc. (Ashton, Bolton, etc.), are the de-
scriptions for the lines, F^, Fg, etc. (under 20s., 20s.-25s., etc.),
the headings of the columns. To any particular sub-group
ABCD Eg Fg (4 loom men cotton-weavers at Stockport,
earning 20s. to 25s.) corresponds one entry (214) in the table.
Of course the sub-divisions by the F's can be omitted for a
simpler tabulation, or a third variable, Gj, Gg, can sometimes
be introduced. In the table given 109 is the total of E^,
799 the total of F^.

It is advisable in many cases to tabulate in three successive
stages: first, the ordered arrangement in full detail of all
the information ; second, the analysis of the first tables under
definite headings as just described ; third, abstract tables of
the main results. The first set are merely for reference, if
minute details may be wanted, or if further analysis may at
some time be needed ; the third set is a mere abbreviation of
the second. In this chapter we deal with the second set.

2. The following table from the Reports on Earnings, etc.,

in the Textile Trades,* will serve to illustrate the discussion.

* Cd. 4545, p. 63.
60

TABULATION

Cotton Industry — Weaving

Number of Men Weavers (4 looms) working full time, whose Net
Earnings in the last pay-week of September 1906 fell witliin the
undermentioned limits.

Under

208. and

26«. and

30s. and

35«. and

Total

Average

20s.

under 25s.

under 30s.

under 35s.

above.

numberi

earnings

Ashton-under-Lyne

109

s. d.
23 10

Bolton ....

—

24 10

Stockport .

214

—

299

23 3

Preston .

406

361

833

24 10

Blackburn

1,293

1,669

121

3,166

25 3

Accrington

190

127

349

24 6

Burnley .

185

1,448

1,942

402

4,063

25 11

Bacup . .

606

203

949

24 4

Rochdale .

258

756

416

1,514

23 4

Other districts . .

122

535

147

811

22 7

Total ....

799

S,SS4

4,989

667

12,123

24 11

Percentages . .

6-5

45-6

41-2

5-5

1-2

100

—

This is an example of double tabulation with cross totals.
The problems isolated for study are the distribution of the
number of weavers according to their earnings, and the
variation of this distribution from district to district. It
forms one of a series of tables in which the variation of wages
according to occupation and district is examined.

3. Before making a table we must consider in detail
exactly what information is wanted. The data generally
consist of one or more items of information about each of
many individual persons or things. In this case we know the
industry, district, occupation, sex, age (whether adult or not),
earnings, and length of time worked, for each person. We
can group any three of these data in a double table. Here
we take as the main heading the composite datum " industry,
occupation, sex, age, and length of time (i.e. full time, 55-^^
hours)," and tabulate according to the remaining two, viz.
district and earnings. We might equally well tabulate

E 2

52 AN ELEMENTARY MANUAL OF STATISTICS

district and occupation, or occupation and earnings, or sex
and earnings, etc. The result of the particular tabulation
used is to show that earnings are nearly uniform district by
district, and are concentrated in the two groups 20s. to 25s.,
25s. to 30s.

Where one of the quantities varies grade by grade (as
wages, age, etc.) it is entered in the horizontal heading. The
number of grades entered separately is limited by the nature
of the material and by the consideration that the whole must
be easily visible at once.

The order of the districts, or other terms, in the vertical
list should be alphabetical if there is no natural order; but it
frequently happens that there is a natural or geographical
g rouping which is of assistance in studying the relationships,
or in making subordinate totals. Similar places or things
should be next each other.*

The line of totals shows the distribution of wages in the
occupation as a whole; the column of totals shows the
distribution of the occupation among the districts.

Supplementary information can be added, if the table is
not overcrowded. A percentage line, as the last, is often
very useful. The column of average earnings makes the
visualization of the figures easier.

In printing, great care is necessary to bring out the
principal words in the heading by sititable type ; and wher-
ever there is a change in the significance of the numbers a
change of type should be made. A typewriter is not capable
of producing a good table.

4. The table should, if possible, show on its face its exact
meaning. It is too commonly the case that a table can only
be understood by a cumbrous system of notes or references,
or by searching through a great deal of preliminary matter.
For this reason the heading is rightly long and carefully
worded. If necessary the heading should be broken up into

* In the table just given the order of places is that used throughout
the report, and is convenient for cross reference. It is partly
geographical, partly according to the nature of the trade in the district.

TABULATION 53

a series of sentences, with great care as to space and typing.
When the matter in hand is extremely comphcated, it is
better to use a brief heading, and to place a full description
of the meaning of the table and definitions of the terms used
in print on the page opposite the table.

It often happens that many of the entries require special
explanation. These may be given by a series of notes legibly
printed immediately under the table. References by *, f, J,
|, fl, etc., should be avoided if possible. Every one has suffered
from the system of notes used in railway time-tables. In the
case before us, the only further definitions wanted are those
of the districts and of the distinction between men and boys.
The former is given by reference to a page where the de-
limitation of districts is stated once for all the tables ; for
the latter one has to search through the introduction to the
report to find that males over 20 years of age are counted as
men.

It is generally the case that, however minute the tabula-
tion, there is a residuum ; here we have " other districts,"
and earnings "35s. and above." The residua should be
made small compared with the total, and should be inserted
to avoid confusion.

After a table is made it is often the case that it has to be
re-cast to fit the printed page. Folded tables should be
avoided ; if the table is too big for a page, or for two pages
facing each other, it should be split up in two or more. The
eye cannot grasp more detail at once than will cover two
pages.

5. A table should neither contain numbers consisting of
many digits nor many blank spaces. The latter can be
avoided by merging the unimportant lines in the residuum.
The former will be avoided if careful attention is paid to the
substance of Chapter II above. Numbers have very seldom
more than a superficial accuracy beyond the third or fourth
significant figures, and it is seldom that greater accuracy is
required, unless for further numerical work. Large numbers
in a table confuse the eye, destroy the legibility of the whole,

54 AN ELEMENTARY MANUAL OP STATISTICS

and conceal the significance of the grouping ; the wood is
hidden by the trees. Either round numbers should be used,
in such a way that the last digit printed is accurate, or the
lines can be given as percentages or per thousands. It is to
be remembered that full details are supposed to exist for
reference in an earlier series of tables (not necessarily
printed).

6. So far we have been considering the form and nature of
tables intended to give public information and resulting from
a collection of statistical data. Tabulation has further im-
portant uses. When an investigation as to any facts is made,
it may happen that the groups, or classes, or series which
result are predetermined in form, and that we have merely to
fill in details in tables already prepared; but it frequently
happens that we are in the position of an explorer, and do
not know even what kind of things we may discover. In
such cases the process of tabulation is the process of analysis.
In the investigation as to wages in the cotton industry, for
example, tables were made to determine how far the number
of looms tended per person influenced wages, what was the
relation between the earnings of spinners and of their piecers,
whether wages were nearly at a uniform level from place to
place, and many other such questions. For analysis of this
kind the rule is simple ; determine exactly what it is that is
to be tested, devise the table that will answer the particular
question and no other, fill in the details from the data, and
perform the necessary arithmetic for any comparison wanted.

7. Again, in considering the progress of an institution or a
business, analysis is constantly wanted, and is carried out in
tabular form. We deal with this subject in Chapter IX.

8. Diagrams, averages and tabulation can all be used for
presenting the results of a statistical accumulation. Of
these the tabulation is the essential. Diagrams only give the
results of tabulations in a special form, suitable for showing
the relations between the various numbers and for allowing a
coup d'csil over the whole field, but they cannot replace the
actual figures for purposes requiring minute accuracy or for

TABULATION 55

further numerical work; also, as stated above, they should
only be used over a limited field, while the tabular form is
universal. Averages are abbreviations, replacing the more
complete table for purposes of comparison with other tables.
The reduction of a column of figures to an average throws
away a great part of the data. Much attention has been
given in recent times to curing this defect of averages, but
after all refinements have been made we cannot dispense
with the details of the group averaged, and these are to be
found in tables.

9. It seemed inexpedient to load this manual with many
examples of tables ; in Part II many small tables are given,
but they should be regarded as the final kind of tabulation,
i.6. "abstract tables of the main results." The reader can
find innumerable examples in statistical publications, and
should criticize them by asking the questions: "Are the
headings intelligible ? Are the terms used in the heading
and the table sufficiently defined ? Is important information
omitted or unimportant included ? Is the spacing and
arrangement of type satisfactory ? Is there any difficulty in
picking out the essential information ? "

CHAPTER VII

SAMPLING

1 . It is not always necessary to obtain complete information
as to all members of a group, in order to give an adequate
account of it. Most practical judgments are formed by
experience of a limited number of examples. Purchases are
frequently made after examination of a sample. The satis-
factoriness of a consignment of goods is tested by examining
and testing a few bars, cases, packages, etc. The probable
yield of a mine is estimated by assaying a small quantity of
ore. The goodness of a water supply is ascertained by
bacteriological examination of a microscopic quantity. Such
methods are not only means of saving time and expense, but
are absolutely necessary in some cases ; for testing often
destroys the commodity, as when a tin is opened or the
breaking-strain of a steel bar is determined, and it is often
impracticable to examine every part, e. g. in the case of a mine,
whose contents is not completely known till it is exhausted.

2. The first essential of an examination by sample is that
every member of the group considered should have nearly
the same chance of being included in the sample. This
may be secured either by mixture or by random selection.
Mixture. — Suppose it to be required to assay the quantity of
gold in several barrels of the sweepings of the Mint, or the
quantity of alcohol in many cases of wine, to take two emi-
nently practical examples. In the first case, extract equal
small quantities of dust from near the top, the middle and the
bottom of each barrel. Mix each sample thoroughly, take an
equal fraction of each and mix (say) four together ; repeat this
process of mixing and division till a quantity small enough to
be assayed is obtained. In all such processes the methods of
choice, mixing and division will be directed to neutralizing
any physical irregularities of weight, shape, etc., which might

SAMPLING 57

destroy the random nature of selection. To determine how
nearly the result is correct, the process should be repeated
(say) four times ; the true result may be expected to be within
the divergencies shown by the four measurements.

3. Random selection. — This is often sufficiently secured by
the process of spreading out the consignment of goods, etc.,
and marking one taken here and another there, avoiding the
first and the last and the most obvious, and testing the
objects marked. Another method is to divide the objects
into equal groups and take one at random from each group.
The more scientific way is to secure absolutely equal chances
by numbering the whole group consecutively, writing down
the numbers on tickets and shuffling them, and finally drawing
at random some of the tickets and examining the objects with
the corresponding numbers. To avoid the writing and drawing,
digits are sometimes selected at random from mathematical
tables and used as if they were numbers drawn at random.

As before, the exactness of the result (if it is a case of
measurement) should be tested bj- repeating the process,
varying the selection each time.

4. In carrying out the above processes successfully in social
or other investigation, less concrete than the examination of
a consignment of goods, the first step is the careful and exact
definition of the group to be tested. If, for example, we are
examining the physical condition of school children, we should
delimitate the area to be taken, enumerate all the schools in
it, and find the number of children on the register of each ;
the group taken would then be co-extensive with the
" registered school children." In making the measurements
we should have to take children absent from school as well
as present, if they happen to be chosen by the selective pro-
cess used, as otherwise we should be taking the smaller group
" children present at school " ; this might give an imperfect
result, as the absent children might contain a large proportion
of the physically unfit. In any case, the group as described
would not contain children removed from the district and
specially treated in institutions.

The temptation is always to measure the obvious and

58 AN ELEMENTARY MANUAL OF STATISTICS

easily accessible ; but if we do this our sample is of " the
accessible," not of the whole group. Thus the budgets of
working-class expenditure, which are often published, are not
typical of the working class as a whole, but of that part of it
which is intelligent enough to have some kind of record and
is willing to communicate private details. In particular, the
expenditure on drink is under-estimated.

5. Determining the average. — It is clear from common-
sense principles that the larger the number included in the
sample measurement, other things being equal, the more
accurately the average will be determined ; in Chapter IV it
was stated that the precision increased as the square root of
the number taken. This accuracy does not depend in any
way on the size of the group from which the sample is selected;
the average height of all the men in England can be
determined with the same accuracy by the same number
of measurements as the average in one town, if in each case
every person has the same chance of inclusion. The follow-
ing examples illustrate the increase of precision as more
samples are included, and other points —

(a) Forty groups of ten entries each were taken at random
from a list of the rate of interest paid by 3,878 Companies.

The average rates obtained for these forty groups were as
follows —

Averages of 10 Compasues selected at random

Rate of Dividend. Number of Occurrences.

£ s.

Above 5

4 18

4 17

4 15

4 14

4 13

4 11

4 10

The average of the 400 Companies, contained in the 40 groups, is
£4 14s. lid ^ ^

The original entries vary from to £103 %. The averages
of 10 are all between £4 10s. and £5 Is. It is then practi-
cally certain that the average of all is between these
limits, and not far from the average of the 40 groups, viz.

SAMPLING

£4 14s. \ld* Actually it is found to be £4 15s. Id., when
all the Companies are included.

(&) A large number of packs of playing cards were mixed
together, and 32 groups of 3 cards were drawn, and the
number of pips on each were counted. Knave, Queen, King
being taken as 0. The following was the result —

Total number of

pips on 3 cards

in order of

drawing.

16
17

14
19
24

10
18

11
15

17
18
14
16

17
5
6

Total Average Total Average Total Average Total and

on per on per on per average for

12 cards, card. 24 cards, card. 48 cards, card. 96 cards.

65 5-4

- 121 5-0

3-.?

3-75

2-9

5-4

3-8

4-2

3-5 ;

. 206 4-29 "I

100

4-2

4-0

196 4-08 )

K02

Average
4-19

* These figures are given and more refined measurements are made in
the Statistical Journal, 1906, pp. 550-3.

60 AN ELEMENTAKY MANUAL OF STATISTICS

The original cards vary from to 10; the averages of 3
from to 8, of 12 from 2-9 to 5-4, of 24 from 3-5 to 5-0. It
is then practically certain from the sample that the average
of all is between (say) 3'5 and 5'0, and that 4'19 is a good
approximation. Actually there are 55 pips to a suit of 13
cards (picture-cards counting blank), and the average is 4'23.

6. While the determination of the average is of great
practical importance for purposes of valuing the group and
other arithmetical work, it is often equally important to deter-
mine the proportion of various kinds in a group, as for example
the number of families per 1,000 whose income is less than£l
per week, or the number of children per 1,000 suffering from
remedial throat complaints. The following examples show
a method that can be followed —

(a) The 400 Companies in the former example were divided
into 4 groups of 100 each and tabulated according to the
rate of dividend paid.

Number of Companies.

Per cent.

Rate of Dividend.

1st

2nd

Srd

tth

Together.

100

Estimate.

Actual.

Nil

£1 and under 3

1-5

£3 „ 4

108

27-2

£4 „ 5

117

29-25

31-1

£5 „ 6

17-7

£6 „ 8

10-8

£8 and above

7-25

5-7

100

400

100

The last column but one shows the distribution as estimated
from the sample of 400 ; the first 4 columns show how far the
estimate can be trusted. Thus it is practically certain that
rather more than half the Companies paid between £3 and
£5, the numbers only varying in the 4 groups from 53 to
59. The number between £1 and £3 is doubtful. The last
line, containing the exceptionally high dividends, is a priori
uncertain ; the accident of sampling may easily include too

SAMPLING 61

many or too few rare cases. The method can only be tnisted
for the large, central divisions. The last column shows the
actual distribution of the 3,878 Companies, from which the
samples were taken.*

(h) In the draw of 91 cards (including all but the last five
of the previous paragraph), the actual occurrence of the
various numbers was —

Ace ... .8

2 . . 81

3 . . 5h20

4 .7}

5 . 7^

6 . 8y23

7 . . 8j

8 4)

9 . 9 [19
10 . . . ej
Knave . 7")
Queen . . . 7l21
King . . 7j

If the drawing had been continued, of course, we should
have found less and less relative difference between the
numbers. Here we have no actual test of the accuracy of
the result.

[A mathematical way of dealing with such a question as
" how many picture-cards are there in the given group ? " is
as follows : Let n be the number in the group, let m be
drawn, and pm prove to be aces. Then pn is the most pro-
bable number of pictures in the pack, but it is as likely as not

to differ from this number by as much as — .\— —. It

^ 3 \ m

is very unlikely to differ by as much as six times the
expression just written.

In the card experiment, if n were 1,820 (the number of
cards actually in the, group used), m = 91, pm = 21. The

* For tie a priori testof accuracy, see again Stat Journal, 1906, p. 553.

62 AN ELEMENTARY MANUAL OF STATISTICS

forecast from the sample as to the number of pictures among
the 1,820 cards would be —

pn + ^ X l^^Q J 21x70 ^ ^ 420 + 54,
^ ^ 3 \91 X 91 X 91 -

and we should feel sure that the number was between
420 + 6 X 54 = 420 + 324.

Per 13 cards we should have (by proportion) 3 + "4,* and
the maximum possible would be about 5.

Thus the experiment is not sufficient to determinate the
proportion of picture-cards accurately. More cards would
need to be drawn till the m in the above formula was
sufficiently increased.]

7. No formal rules can replace judgment and experience
in the selection and interpretation of samples. The simplest
practical direction is to continue to increase the number of
samples, till successive tests show sufficiently similar results.
When dealing fi^equently with the same kind, of course,
experience would soon show how many tests were sufficient.

8. Two other methods of sample measurement are some-
times used.

Suppose we wish to test the knowledge of a large class of
students (say 100). We might by some very simple examina-
tion, or by consulting the teacher, place them roughly in
order of intelligence, and then examine in detail, say Nos.
1, 10, 25, 50, 75, 90, 100 (the maximum and minimum,
median, quartiles and two deciles)."|' Thus a good estimate
could quickly be obtained, and the relative ability of two
similar classes quickly judged.

In the same way we could describe any group that can be
placed in order, by the detailed examination of a few selected
hy rule. This method differs essentially from the method of
random selection already explained.

* Observe that this result is independent of n.

t The deciles are the values which divide a group into ten equal parts,
in the same way as the quartiles divide it into 4. If these seven positions
are determined quantitatively for any group, a diagram of the form A,
p. 37, can he drawn with considerable accuracy.

SAMPLING 63

9. Kather than trust to the arbitrary action of chance,
some investigators prefer to choose what they believe to be
typical groups, and examine them in detail. Thus, investi-
gations as to the wages, etc., of agricultural labourers have
been conducted by selecting some forty districts through-
out the country, so as to include types of all kinds of
agriculture, and of all economic situations. This method
results in an accurate and intelligible picture, but there is no
easy means of calculating any average, or of knowing the
distribution by number of persons earning various rates of
wages. For filling in details where the general results are
known the method is to be recommended.

CHAPTER VIII

RULES FOR USING PUBLISHED STATISTICS

1. It is never safe to take published statistics at their face
value, without knowing their meaning and limitations, and it
is always necessary to criticize arguments that are based on
them, unless one is able to trust implicitly the knowledge
and good faith of the persons bringing them forward. It is
extremely easy to falsify the lessons which numerical state-
ments should teach. The actual use and appreciation of
statistics is ultimately a matter of intelligence, special know-
ledge and common-sense; but the following nine rules
suggest the lines of study and criticism.

First. — Find the exact definition of the units which go to
make the total. What is a soldier ? What one pound's
worth of exports ? What a registered birth ? What a
member of the population, a case of fever, a bushel of
wheat ? One of the standard questions in agricultural
statistics is " What is a cow ? " In every case the definition
depends on the regulations and method of collection. Thus
we need to know at what stage a recruit is entered as " on
the strength of the regiment " ; what goods are counted as
exports, and how they are valued; whether all births are
registered, and whether still-births are included ; how
travellers, absentees and the homeless are counted; what
are the rules for diagnosis of fever ; whether wheat is weighed
or measured ; when a heifer grows into a cow, and much
more detail of this sort. Generally expert knowledge is
needed; sometimes the report on the statistics contains
sufficient explanation and definition; sometimes the whole
can be worked out from a study of the blank forms of

RULES FOR USING PUBLISHED STATISTICS 65

inquiry (with instructions) on which the original data are
obtained.

The apparent meaning of a total is seldom its real mean-
ing, but generally results from an artificial definition,
necessitated by the process of collection.

As examples may be suggested the discovery of what is
meant (i) by a room, (ii) by a farmer, in the census
reports.

2. Second. — Consider how far the persons or things grouped
together in a total or sub-total are similar; in other words,
how far the group is homogeneous. Thus, persons whose
occupations are grouped under the main heading "Textile
Fabrics" differ with respect to (1) sex, (2) age, (3) nature of
the material worked (cotton, wool, etc.), (4) position in the
industry as merchant, dealer, manufacturer, or employe, (5)
specific occupation, (6) locality. If we are merely told that
1,155,397 persons were included under the main heading in
England and Wales in 1901, the information is so wide as to
be nearly useless. An example of the most minutely defined
group given in the census reports is : — County Borough of
Oldham : number of males between the ages of 25 and 45
engaged in spinning process in cotton was 2,711. To know
the meaning of this we should have to go carefully through
a spinning-mill.

Whether the group or sub-group is sufiiciently homo-
geneous depends entirely on the purpose for which the figures
are used. If we compare the total numbers in the cotton
industry in 1891 and 1901 we should be misled, because the
numbers of children, men and women are in quite different
proportions at the two dates ; but a useful comparison might
be made between the numbers of men.

The possibility of change in the relations shown when the
groups are analyzed into parts of greater homogeneity must
always be borne in mind. Innumerable examples might be
given; an important one arises from death-rates. The rate
is calculated by dividing the number of deaths in a district
in a year by the number of persons living in the district

66 AN ELEMENTARY MANUAL OF STATISTICS

midway through the year, and multiplying by 1,000. Analysis
at once shows that the various age-groups of the population
are subject to quite different risks of death, and that the risks
differ also according to sex; further, deaths from accident,
from infectious diseases and from other causes should be in
different categories.

If the internal constitutions of two groups are the same,
e.g. if the distribution by age and sex are the same, then
averages based on them may be properly compared. But we
must never assume either homogeneity or similarity of
division without knowledge.

3. Third. — Having defined and analyzed the totals, the
next question is what is the relation of the quantity they
measure to the quantity as to which we want knowledge.
We wish to know the stress of unemployment, we learn the
number of trade-unionists out of work; or of poverty, and we
are told the number in receipt of public relief; or we are
examining the improvement in health of the population, and
we find the amount of disease and the number of deaths ; for
education we can tell the number of students, or of student-
hours, or of examination successes. These statistical totals
and averages are at best indices, not actual measurements,
of the more subtle and often incommensurable quantity or
quality, which is essentially the object of the investigation.
In order that indices may be useful they must at least move
up and down with the quantity they represent, as the ther-
mometer moves with heat and the barometer with pressure,
and they should further make great or small oscillations
with great or small movements ; but many of them have less
relation to the complete phenomena than the thermometer
has to sensation of heat (which depends also on moisture and
physiological conditions), and may be as remotely connected
as the fall of the index of a barometer with the fall of rain.

If experience shows that the indices are sensitive and
trustworthy they may be used to bridge over the gap between
one more complete measurement and the next.

4. Fourth. — Before trusting or even reading a statistical

RULES FOR USING PUBLISHED STATISTICS 67

account, it is well to sit down and think quietly what
statistics ought to have been collected, if possible, for the
purpose in hand, and what sources of information exist, or
should exist. Thus, if wages were to be measured, we should
decide that the weekly rate, the annual earnings allowing for
unemployment, supplementary earnings and the earnings of
other members of a man's family should be known, and that
allowance should be made for any necessary expenses ; further,
the money value should be interpreted in purchasing power,
and the standard of life attained should be clearly shown. Of
these things some it is not possible to measure ; we cannot
measure the actual satisfaction obtained from the expenditure
of money, nor the value of unpaid personal work. Others, as
the annual receipts and complete expenditure, could only be
measured if the persons concerned kept accurate accounts.

Having got so far, we may take up the statistical report
and consider how far the problem has been understood,
whether all the practicable measurements have been made,
and whether the result gives a true index in the sense of
the last paragraph. We can thus decide as to whether the
information is sufficient for solving an}' assigned problem ; in
only too many cases we find that it is not.

Further, if there is any suspicion of bias, of the intention
to support any preconceived view, the criticism of method
must be particularly rigid, and the maximum possible effect
of the unconsidered factors must be allowed for.

5. Fifth. — When we have to deal with averages, rates, and
percentages, we must carry our second rule of criticism
further. Not only must we consider whether the numerators
and denominators are homogeneous in themselves, but
whether the terms of the denominator have a reasonable
relation to those of the numerator. Should, for example, the
number of deaths from small-pox be counted in relation to
the whole population, to the vaccinated population, or to the
number who contract the disease ? Should the birth-rate be
reckoned per 1,000 of the population or per 1,000 of married
couples ? Should the production of coal per head be reckoned

F 2

68 AN ELEMENTARY MANUAL OF STATISTICS

with respect to the population of a nation, or to those engaged
in the coal trade, or only to the coal-hewers ? The general
answer is that the denominator should be limited to those
who have a direct relation to the numerator; the legitimate
birth-rate, e.g. should be in relation to married couples with
some restriction of age. It may happen that this restricted
denominator has a constant relation to a larger population,
and in that case the latter may be used for simplicity of
working (sometimes for lack of the detailed information), and
for comparison with similar averages. Thus the number of
births (929,807) in 1901 in England and Wales may be stated
as 160 per 1,000 married women or as 28"3per 1,000 persons ;
this last results from the combination of the two rates, 160
births per 1,000 married women and 177 married women per
1,000 persons. If the 177 remained unchanged, the two rates
160 and 28"5 would of course have a constant ratio to each
other.

6. Sixth. — When two quantities are compared we must
consider whether they are strictly comparable, and for this
purpose most of the foregoing rules are necessary. Com-
parisons are made between two similar measurements at
different dates («. g. population, death-rate, average wage,
production of wheat, etc.), or between two similar measure-
ments relating to different places (e.g. trade, consumption of
meat or wheat per head, amount of taxation per head, total or
average income in two countries, etc.). We must test whether
the two measurements are made on the same basis, so as to be
indices of the same kind of phenomena considered, so as to
cover the same ground and suffer from similar "error of
bias " (see pp. 29, 32). Having ascertained this and so used
rules 1 and 3, we then apply rules 2, 4, and 5 if necessary.

By such means we shall readily realize the difficulty of
minute comparisons over long periods, during which relations
have continually changed, and the extreme roughness of
comparisons between such measurements as the indices of
prosperity of two nations. Accurate comparisons can only be
made between closely similar things or over quite short periods.

RULES FOR USING PUBLISHED STATISTICS 69

7. Seventh. — Closely connected with the last is the measure-
ment of accuracy. In Chapters II and IV the approximate
nature of statistical measurement was discussed, and some
methods were given of testing the accuracy of results. In all
statistics we must decide whether the data and methods will
yield results accurate enough for the arguments based on
them. It would be absurd to speak of an increase in average
wages from 20s. 3(^. to 20s. Qd. in twelve years, for the
average could not be determined to Id. in either case, and the
group considered would have changed its character in the
period ; but we could speak reasonably of an increase of
" about 50 °/ " if the averages were 20s. and 30s. The less

/o o

the groups satisfy the stringent conditions of the first six
rules laid down, the greater must be the margin allowed for
error. Where possible, the greatest possible errors arising
from imperfection of data or processes should be worked out.

8. Eighth. — We must not depend on figures relating to
single days, months, or years, or on comparisons relating to
short isolated periods. In Chapter V the fact that every
measurable recurring phenomenon yields a series of definite
characteristics was illustrated. These characteristics, the
natures of the fluctuations and of the trend, must be known.
In the case of the population of a large country, where there
is little emigration or immigration, it is not difficult to fill in
estimates for intermediate years ; in the case of the total value
of exported goods it is impossible. Every measurement must
be viewed in the light given by a series of similar measure-
ments stretching back over a long period ; otherwise temporary
fluctuations will be taken for permanent changes, as if a cold
summer were regarded as proving a change in climate ; or a
rise will be reported, when the whole trend is downwards, as
if we should compare the bank holiday traffic of a decadent
tramway one year with the lowest day's record of the
preceding.

Where a sufficient record cannot be obtained, judgment
must be suspended.

9. Ninth. — Having determined as far as possible the exact

70 AN ELEMENTARY MANUAL OF STATISTICS

purport and limitations of the statistics, consider (without
reference to the printed report) to what conclusions they lead,
or whether they are so imperfect that no conclusions can be
reached without further investigation. There is often a great
gap between the statistical table and the non-statistical con-
clusions that are fathered on to it, especially if the statistics
were obtained in order to support a preconceived theory.
Statistical work properly ends with such a dull, colourless,
matter-of-fact report as is customary in the publications of the
British Government. As a separate process such results are
to be taken in conjunction with non-statistical knowledge.
Inferences are suggested and tested by the reported facts,
and a severely critical and logical analysis is necessary before
the whole investigation leads on to some reasoned action.

CHAPTER IX

METHODS OF STATISTICAL ANALYSIS

1. In the previous chapter the way to criticize statistical
reports was outlined ; in this chapter we consider hriefly the
methods of collecting statistics at first hand, (i) for the
purpose of testing the progress of a commercial undertaking,
(ii) for testing the success of an institution, (iii) for collect-
ing data for the solution of a social problem.

2. Details vary so greatly for different kinds of business,
that it is only possible to lay down some general principles
with illustrations. The processes of book-keeping and ac-
countancy are, in their more refined forms, examples of
statistical investigation, and, so far as £ s. d. is concerned, pro-
vide the data, even if they do not give the result, of such
analysis. When accountancy is applied to commodities as
well as to money, we arrive at statistics. Take the case of
wool-spinning. The data that should be tabulated are —
the weight and cost of raw wool used in a given time, in
the aggregate, in each room, and by each mule ; the weight
of yarn produced, in similar detail, and the weight of waste
material recovered ; the price realized for the products (or, if
the yarn is used in the same factory, the estimated value) ;
and the cost in wages and in oil and sundries. Over a
longer period an estimate should be made for the interest on
the capital value and the depreciation of the machinery used,
together with a proportional allowance for the general ex-
penses of the factory, such as salaries, rent and rates, and
advertising. The cost of the engine should be placed under
the special expenses, if possible ; if not, this cost must be
divided between the various rooms with what accuracy is
practicable. With such data it is possible to tell what

72 AN ELEMENTARY MANUAL OF STATISTICS

machines, rooms, or departments are running at a loss, or
just paying their special expenses, or contributing adequately
to the general expenses, or making a profit.

In this case, also, it is easy to state the number of lbs. of
wool spun and the length of the yarn produced, and the
actual work done by each group of operatives (the spinner
and piecers at each pair of mules), which is, in fact, measured
for the basis of piece-wages. It can be at once determined
whether the machine (the spinning-mule) is being used
efficiently.

Similarly in weaving, data are easily available for the pro-
duct per loom, per operative, and per £ of wages paid, and
the totals can be made for each weaving-shed and for the
factory as a whole.

3. A more complicated problem is presented in railway
working, and an example of the method of compiling statistics
in use on many important railways in America, India and
elsewhere is very instructive. The data are twofold, based
respectively on the details of the train service, and on the
quantity of goods conveyed ; the first are connected with
expense, the second with remunerative work done.

For each journey of each train the guard sends in a report
as to the time at which the engine arrived, the times (actual
and due) at which the train arrived at and left each stopping
point, and as to the number of wagons (empty or loaded)
hauled each section of the journey, with other details. For
each journey of each engine the driver reports the time
he was working with the engine, and how it was allotted to
standing, shunting, or running, the amount of coal taken on,
and the number of the wagons hauled in each section of the
route.

On the other side, returns are made of all consignments of
goods, showing the stations at which they were received and
where they were delivered, and the money received for their
transit.

From these data the following tables among others are
compiled : —

METHODS OP STATISTICAL ANALYSIS 73

Statistics of Operation. Goods and Minerals together.
Months of June 1904 and 1905.

No. of
work-
ing-
days.

Tons carried.

Ton-
miles.

Average
train-
load.

Train-
miles.

Average distance hauleii,

Total.

Per
work-
ing-day.

OOO's
172
166'

Goods.

Minerals.

Together.

1904
1905

26
26

OOO'a
4468
4325

,00000's

1026
980

Tons
100-3
;i04-5

OOO's

1023
938

Miles
35-4
34-3

Miles
18-5
18'6

Miles
23-0
22-7

£ngine-hoiirs.

Ton-miles
per engine-
hour.

Train-miles
per train
engine-
hour.

Train-miles
per single

track-mile
per work-
ing-day.

Ton-miles

per route-
mile per

Train.

Shunting.

Total.

working-
day.

1904
1905

OO's
1386
1263

OO's
1922
1821

OOO's
331
308

310
318

7-38
7-43

13-2
11-8

2363
2223

Wagon-miles,

Average train-load
of wagons.

Wagron-miles
per engine-
hour.

Ml*

fH J-

%
Loaded of

■g

■■a .a

"Sj

Total.

-1

oj a

•5

0000' s

OOOO's

Tons.

1904

1939

1174

3113

62-3

18-9

11-5

30-4

224

162

94-1

5-29

1905

1821

1123

2944

61-9

19-4

12-0

31-4 233

162

95-4

5-38

" Ton-miles " form the principal measurement of the
revenue-yielding work done by a railway so far as freight
is concerned, and are obtained by multiplying the number
of tons in each consignment by the number of miles it is
carried. " Train-miles " signifies the aggregate of the miles
run by trains; "engine-hours" the aggregate of the hours
in which an engine was working with a train, distinguishing
running from shunting. The total of wagon-miles is com-
puted by multiplying the number of wagons moved by the
number of miles run separately for every section at the
beginning of which the composition of a train was altered,

74 AN ELEMENTARY MANUAL OF STATISTICS

and adding these products. These results, together with the

total of the tons moved and the fixed information as to the

track, are sufficient for the tables.

Let T be number of tons moved, T„^ number of ton-miles,

T^„ number of train-miles, E, and E^ numbers of train and

shunting engine-hours, W, and W^ the number of loaded and

T
empty wagon-miles. Then the average train-load is ~

tons ; the average distance hauled is y^ ; the average of ton-

miles per engine-hour is — — ^ ; of train-miles per train

E( + Es

engine-hour is -^ ; the average train-load is — '-^ «

T
wagons ; the average wagon-load is =™> and the average num-
ber of wagon-miles per engine-hour is '^ '■

Ej + Ej

Such figures could be worked out for any division of the
railway that is required. By comparing the averages obtained
for different months or different divisions, we can observe the
work done by engines in hauling goods or wagons (ton-miles
or wagon-miles per engine-hour), the use made of the track
(or railway as ordinarily understood) and of double lines
(train-miles per track-mile and ton-miles per route-mile),
what proportion of haulage is effectively spent in hauling full
wagons, and how heavily the wagons are loaded. Where
any one of these averages increases, there is presumptive
evidence of growing efficiency in working ; where a
difference or decrease is shown, there is a case for inquiry
as to the cause ; it may prove to be due either to the nature
of the work, or to incompetency in handling it, or to a
reorganization which produces a compensatory improvement
elsewhere.

From similar tables the receipts per ton, per ton-mile and
per train-mile are worked out for different classes of traffic,
and brought into close relation to the cost.

METHODS OF STATISTICAL ANALYSIS 75

4. In the case of railways and other large undertakings
the problem is to discover exactly what measurement is most
sensitive to efficiency of work, and to devise the necessary
machinery for obtaining the statistics of precisely that
measurement. In the running of goods trains the principal
expense that can be reduced is the time during which the
wages of the three men (driver, fireman and guard) concerned
are paid ; " wagon-miles per engine-hour " and " ton-miles
per engine-hour'' are found to provide precisely the tests
wanted. In other cases it might prove to be the production
per spindle per week, or the output of coal per hewer. When
such tests are devised and kept systematically, an instant
indication is given of any improvement or slackening in the
work, and the reasons of the change can then be investigated.

5. It is clear that such a broad average as " wagon-loads "
obtained by dividing 103 million ton-miles by 20 million
loaded- wagon-miles does not satisfy the test of homogeneity
suggested above (p. 65); a railway may be engaged in
hauling coal by the train-load and also in handling small
parcels for quick delivery ; for the former heavy wagon-loads
are easily obtained, with the latter the rapidity (and the
custom) may be lost if goods are not forwarded till a wagon-
load is ready. In other industries high average production
may depend on inferiority of goods. Where the relative pro-
portions of the different classes of work done vary very little,
this consideration will not vitiate the comparison of averages ;
but where the proportions are not steady, further analysis
and subdivision must be made, so far as practicable, till
statistics are obtained for nearly homogeneous work; the
first step made in this direction in railway statistics is in
separating minerals from other goods. In the same way the
analysis should extend, both for quantity and cost, to the
smallest subdivision of the work that can be separated.

The labour and expense of collecting statistics in this way
is much diminished if, when the actual averages or quantities
which form the most delicate tests of efficiency have been
decided on, no statistics are accumulated, which are not

76 AN ELEMENTARY MANUAL OF STATISTICS

directly needed for these averages, etc., and simple printed
forms are used, which can be easily filled in an ordinary
routine; these forms should be regularly delivered to a
statistical clerk, who should systematically tabulate them on
a uniform scheme.

6. There are two considerations which affect the use and
formation of such statistics; first, the value of money is
subject to continuous changes ; secondly, it is not easy to find
a common measure of the work done.

For the change in the value of money the reader is referred
to Part II, Chapter IV, below, with the suggestion that
special index-numbers should be formed to suit particular
circumstances.

The addition and comparison of unlike quantities can often
be made by the device of " weighted totals." This can be
illustrated by the general statistics of the worsted trade.

Export op Worsted Tissues

1894.
Yards.

1907.
Yards.

Mean
price,
1894-
1907.
ti.peryd.

Numl)ers ad-

j usted to common

measure.

1894.

1907.

Broad coatings, all-wool
„ „ mixed

Narrow coatings, all-wool
„ ,, mixed

Stuffs, all-wool .
„ mixed

OOOO's

H17

385

217

272

1320

7756

OOOO's

1379

720

159

104.3

6559

46-2
28-0
31-7
20-3
11-9
9-4

Total ofWorsted Tissues 11067 9901
Ratio . . . 100 : 89-4

Total value . £6,666,000 £7,394,000

Ratio . . . 100 : 1109

OOOOO'a
2234
462
304
245
660
3102

7007
100

OOOOO'S

2758

864

143

521

2624

6967
: 99-4

During this period (1894-1907) the price of wool and of
woven tissues fluctuated considerably, 1907 being a year of
high prices. The aggregate value is therefore not a fair
measure for comparison. The value in four of the six cate-
gories into which the exports are divided fell, and rose in the
other two. The aggregate yardage fell. Now, a yard of

METHODS OF STATISTICAL ANALYSIS 77

" Broad pure wool coatings '' cannot properly be added to a
yard of "mixed stuff"; the first is much heavier, broader and
more expensive than the latter. The average prices of these
six classes are shown in the table ; the first is worth five times
as much as the last per yard. Assume that these prices are
proportional to the intrinsic values of (or to the work done
in producing) the cloth, and for simplicity of computation
take integers nearly in the proportion shown. These are
called " weights " in the sense of Chapter III above, and it
is shown (pp. 18, 32) that they need not be taken with gi-eat
accuracy for purposes of comparison. Multiply the quan-
tities by the "weights," and so obtain the last two columns;
here in effect the unit is "one quarter of a yard of mixed
stuff" equal to ^V^^ o^ ^ J^^d of pure broad coatings, etc.

The comparison of the weighted totals shows that the
total production was practically the same in 1894 and 1907
on this basis, though the value rose 11 % and the aggregate
yardage fell 11 %.

The actual average weights of wool per yard in the various
classes might be used for the statistical weights if they could
be estimated. This method is used in the railway statistics
of live stock, when one horse is counted as equivalent to so
many sheep or to so many fowls, for purposes of transit cost,
and it is capable of wide and varied application.

7. It is often useful to make and keep up to date charts
of prices, cost, output, wages, etc., in considerable detail. In
particular, if a trade is seasonal, it is well to have a graphic
record of the seasonal fluctuations, with a view to forecasting
the immediate future, and to providing an adequate supply
for the probable demand.

It is generally interesting and sometimes of importance to
preserve a record of the rates of wages paid to various classes
of operatives, and also the average for the whole. It has
frequently proved to be the case that the average has risen
faster than the rates, owing to the different growths of
various grades of labour and to readjustments of work. Such
changes are often unobserved, but are frequently the main

78 AN ELEMENTARY MANUAL OF STATISTICS

factors in the growth (or, less frequently, the diminution) of
earnings.

8. The principles of measuring the progress and efficiency
of an institution are similar to those just outlined, but the
statistical aspect is less important ; for, while a commercial
company is in business for the dollars and the test of success
is pecuniary, an institution exists for carrying out some
defined aim, for -which there is in general no numerical
measurement. Nevertheless it is more necessary to test
the statistics offered by the management of an institution,
especially when it is appealing for help, than those collected
by a commercial body for itself; for it is in the interest of
the latter to know the facts exactly, while the former needs
to show a good case, and there is nothing so easy as to show
a biassed result without actually falsifying the facts. In its
own interest, for success in working, an institution should
record its facts on a commercial basis, and in candour should
present these records to the section of the public concerned.
Hospitals, asylums, schools, colleges, and propagand ist, religious,
philanthropic and social societies are among the institutions
to which these remarks apply.

As regards £ s. d., accounts should be kept in great detail
and carefully allotted to services and departments. In
particular the expenses of advertising, of collecting money,
of printing, of postage, and of administration, should be
shown clearly, and separated from the expenditure directly
on the objects for which the institution exists; the former
correspond to the general expenses of manufacture. Further,
when building, new or old, is involved, the exact state of the
building account should be shown, and the amount spent on
rent, interest, rates and taxes. When these things cannot
be found clearly in a balance-sheet, suspicion always may
arise that there is something to conceal. The proportions of
the foregoing expenditures to total expenditure afford tests
of the efficiency of administration that, when applied with
knowledge of what has been done in similar cases, are very
useful.

METHODS OF STATISTICAL ANALYSIS 79

The costs of carrying out the objects of the institution
should then be allotted, so far as they can be properly
credited, to a department or group of departments. Averages
should then be worked out- — for a hospital, the cost of food
and other household expenditure per head per week ; for a
school, the cost of teaching per child per term ; and so on.
At this point the question of homogeneity must be con-
sidered. The averages just mentioned would be useful in
an asylum or workhouse or general hospital usually nearly
full, and in a large primary school, but not in an institution
where there were many grades of expense, or a college where
there was specialized teaching for small classes ; nor would
one judge a missionary society by its expenditure per convert.
The less an institution belongs to a regular type, and the less
uniform the persons it deals with, the less, also, can general
averages be usefully applied ; but where it is possible to
compare like with like, then the causes of differences in such
averages should be sought out.

9. As regards statistics of results, of success in carrying
out the declared aim, it is well to apply Rule 4 of the previous
chapter; think out what is the exact measurement that is
wanted. In a hospital the number of patients dealt with,
together with the average length of stay,* and details of the
number cured or relieved, should be known ; the number of
operations is often stated, but it may include the extraction
of a tooth as equal to tracheotomy, and is not of much use.
In an asylum the number of persons should be given
classified by sex, age and length of sojourn. In a teaching
institution the difficulties are greater. No sensible person
regards examination tests as adequate. The number of
registered students is misleading, as the amount of time
nominally given and the regularity of attendance vary
greatly; the information should rather be given in detail,
showing (for example) the numbers of students, subdivided
by age and standard of instruction, the number of classes

* A railway statistician would probably ask for "patient-days per
bed."

80 AN ELEMENTARY MANUAL OF STATISTICS

per week attended, and a measure of the regularity of
attendance; also the size of the classes should be stated.
In some cases total teaching-hours, total student-hours, and
student-hours per teacher may be stated with advantage ; but
these are likely to be misleading and suggest resemblance
between railways and the business of teaching, which would
only be found in a very wooden educational scheme.

A more useful way of studying such statistics is to com-
pare them in detail year by year, and to try to account for
the differences shown, remembering that the smaller the
numbers dealt with the more apparent will be the variation
from causes that are fortuitous and independent of the
management of the institution.

In the end, statistics of this kind can only help to form
judgments, which should be based mainly on non -statistical
observation.

10. Our final subject in this chapter is the collection of
data in connection with some social inquiry — for example,
the amount of unemployment, the physique of children, the
condition of a district as to overcrowding, or the more
elaborate investigations that have been made as to general
social conditions in London, York, Dundee, West Ham and
Birmingham. The first thing to do is to think out in a
quiet hour exactly what we desire to know, and, next, what
part of this knowledge can rest on a statistical basis. For
unemployment we might decide that the essential thing to
discover was the number of hours' work obtained in the
previous month, for overcrowding the number of cubic feet
in a tenemect per occupant, and so on, but we should at once
find that additional measurements were necessary — e.g. in
the last case the ages and sex of the occupants and the
condition of ventilation. At this stage it is best to work
out blank tabulations, where each column, row and total
worild give definite information on the subject of inquiry;
then work out forms of questions, the answers to which
would lead to the tabulation desired. Next consider what
persons possess the information required.

METHODS OF STATISTICAL ANALYSIS 81

The construction of the blank form of inquiry on which
the answers are to be entered depends on the education and
position of the people who are to fill it in. In general, it is
useless to issue blank circulars unless the filling them in is
compulsory. If the information already exists in written
form, e.g. the record of wages paid at a factory, it can fre-
quently be obtained by a personal visit at which the object
of the inquiry is briefly explained, and interest aroused or at
any rate consent obtained ; and then a blank schedule carry-
ing a clear explanation of what is wanted on its face, can be
left. The questions must be such as can be answered by
"yes" or "no" or in numbers; adjectives such as "fair,"
'■ occasional," etc., are nearly useless for tabulation, their
significance varies from person to person. If, on the other
hand, the data must be collected first hand, a house-to-house
visit may be necessary. The labour may be abbreviated if
the method of samples (Chapter VII above) can be strictly
applied. Of course, tact and experience is necessary for this
work. A separate blank form, again containing perfectly
definite questions, should be used for each case ; but except
where measurements are necessary, the answers should be
obtained in conversation and entered immediately after-
wards; for a visitor taking notes is likely to be an object
of suspicion.

11. The data having been collected, their working-up can
be done in the light of the previous chapters. The special
difficulty in this kind of investigation is the essential in-
definiteness of the quantities (poverty, physique, etc.) to be
measured. It is well not to draw a single definite line, and
say above this line is health, below it weakness, or above this
mark competence, below poverty, but to remember that health,
poverty, unemployment, overcrowding, etc., are relative. The
final statistical table should be a graduation— so many tene-
ments where there was more than 500 cubic feet per
person,* so many at 400 to 500 cubic feet, and so on. Then

* In this case a person should mean an adult, and children should be
counted as fractions according to their age.
o

82 AN ELEMENTARY MANUAL OF STATISTICS

the effect of drawing the line at various grades can be
observed.

All statistics which cannot bear full criticism should be
put aside, even if the inquiry has to be given up ; imperfect
statistics on such questions are often only productive of
harm. In publication, the whole method of inquiry should
be clearly and frankly shown, the tabulations should be per-
fectly clear, and the statistics of the inquiry be definitely
separated from other parts, which deal (for example) with
supposed causes and suggested remedies. Space should not be
wasted in printing elaborate tables of data, but enough detail
must be shown to allow a critic to form an accurate judgment
as to the adequacy of the inquiry.

PART II

G 2

CHAPTER I

THE POPULATION CENSUS

1. The population of the United Kingdom has been
counted once in ten years ; the first census was in 1801,
the most recent is that of 1901 ; midnight before the first
Monday in April has been the date taken in recent censuses.
Blank forms are left with every householder, whose duty it is
to enter certain particulars as to every person dwelling in the
house alive at midnight. Precautions are taken to avoid
omissions and duplications, and persons not in houses are
counted as far as possible. A supplementary test of popu-
lation is afforded by the enumeration of the number of
inhabited houses.

The population enumerated for a district is thus the
number who happened to be there at a particular moment,
which differs from the number who live there habitually and
differs greatly in many important cases from the number who
work there. The accidental element arising from absence on
journeys or presence on visits is not important in most cases ;
but it is evident that the population of holiday resorts fluctu-
ates greatly through the year, and that the selection of April
is arbitrary.

The principal questions put to every person are as to age,
sex, condition as to marriage (known as " civil condition "),
occupation, and birth-place. The number of rooms occupied
by the family group is stated, if less than five. The Royal
Statistical Society is continually pressing for a more frequent
census and for improvements in, and additions to, the questions
asked.* Readers should study the provisions of the Bill re-
* See Statistical Journal, 1908, pp. 496-8, and 1909, pp. 574-593,

86 AN ELEMENTARY MANUAL OF STATISTICS

lating to the Census of 1911, and follow the movement in
favour of a permanent Census OfSce and more adequate
information.

The organization of the census and the working out
and publication of the results are entrusted to the three
Kegistrar-Generals of England and Wales, Scotland and
Ireland. The forms of questions and the methods of publica-
tion differ in the three countries. The principal general
results for the United Kingdom are brought together in the
General Report on the Census for England and Wales.*

Preliminary reports stating the population and number of
houses, in all the subdivisions dealt with by the census larger
than the parishes, are published within a few weeks of the
date of taking the census. This was followed for England
and Wales during 1901-3 by a series of volumes dealing
with one county at a time f ; next came summary tables, J
bringing the figures for the various counties and districts
together, dated April 1903, and finally, and not till July
1904, the General Report comparing the results and methods
with those of the previous censuses, and containing important
tables relating to special questions.

The Scotch Census for 1901 was published in three volumes;
the first, dated July 1902, dealt with Houses and Population ;
the second (February 1903), with Ages, Birth-places, Civil
Condition and Education; and the third (August 1903), was
devoted to Occupations.

For Ireland, separate parts are issued for each county,
separate volumes for each of the four provinces, and a general
report for the country; for the last census these were all
published before the end of 1902. The general report shows
the size of agricultural holdings in considerable detail.

We shall presumably be without any general occupational
statistics relating to the United Kingdom, more recent than

* Cd. 2174. Price 2s. 8d. Pp. 302-315.
f There Ib also a volume relating to islands in the British seas,
i Cd. 1523. Price 2s. 6i. This is the volume most useful for
general reference.

THE POPULATION CENSUS 87

those of April 1901, till September 1913, and the great prac-
tical reforms which are contemplated as to unemployment
and training will remain without adequate statistical basis.

2. There is very great confusion and complication between
the various subdivisions * of the United Kingdom, and the
boundaries are very frequently re-adjusted. The main tabu-
lation of the census relates to " registration counties," of
which there are forty-two (including London) in England;
Wales is divided in three, Monmouthshire, South Wales and
North Wales. Each registration county is divided into districts
(Lancashire, for example, contains 27), and these into sub-
districts.

For administrative purposes England and Wales is other-
wise divided into County Boroughs and Administrative
Counties, and the latter into Municipal Boroughs, other
Urban Districts and Rural Districts, and these into civil
parishes. It is likewise divided into Parliamentary Coun-
ties and Boroughs, which form or are subdivided into con-
stituencies. A further general division is into Ecclesiastical
Provinces, Dioceses and parishes, and another into County
Court circuits and districts. Each of these four methods of
division is subject to continual alteration in detail, as the
shifting of population requires.

To make confusion worse confounded there is also the
division into ancient counties, as given in geography books,
where London is cut up into parts of Middlesex, Surrey
and Kent.

The census publications deal with all these divisions, record
is preserved of all changes, and the general rule is adopted
that for comparative purposes the population on the area as
defined in the most recent census is stated alongside with
the population on the same area (however it was formerly
divided) in previous times. With each county volume is
issued a map, showing as plainly as possible the separate
divisions.

3. To make any detailed comparison for a district it is
* gee note at end of ctapter,

88 AN ELEMENTARY MANUAL OF STATISTICS

necessary to work carefully at the county volume. For ex-
ample, consider the question, "In what way did the district
commonly known as Hull, with its suburbs, change its cha-
racter between 1891 and 1901?" The population of the
present area of the County Borough of Kingston-upon-Hull is
given for both years, but the boundary was slightly changed
in 1897, and in any case is not coincident with the area
affected by the neighbourhood of the great town. " The Regis-
tration District of Hull " (No. 521, in the county of York, East
Riding), " is co-extensive with the civil parish of Holy Trinity
and St. Mary.'' This district is divided into Humber and
Myton, the boundaries of each of which were changed between
1891 and 1901. Next to Hull is the Registration District of
Sculcoates, divided into eight sub-districts, seven of which
contain parts of the County Borough of Kingston-upon-Hull,
which latter contains the whole of the Registration District
of Hull. The sub-districts are further divided into parishes
and there are many illuminating footnotes, such as "A con-
siderable proportion of the population of Willerby Civil
Parish was enumerated in Hull City Lunatic Asylum." On
looking at the map we see that Sculcoates (No. 520), contains
the purely rural division of Ferriby (population 1,845), and
the Municipal Borough of Hedon (population 1,010).

For persons who do not know the district, the only method
is to himp the two registration districts of Hull and Sculcoates
together, subtract all the small districts named where the
population is less than (say) two to the acre, and regard the
residue as the population of what is generally known as Hull.
This will give an adequate idea of the growth between 1891
and 1901, and for a single year can be made correct to within
5 per cent. Such estimates should, of course, be given in
round numbers. Persons in the district, on the other hand,
will find that the information published is intelligible and
sufficient.

4. The preceding paragraph shows that the problem of
tracing the growth of urban population is not to be under-
taken by the beginner in statistics. As a matter of fact it

THE POPULATION CENSUS

is almost impossible to define the difierence between urban
and rural population, except by the technical and almost
accidental division into " urban districts " and " rural dis-
tricts," according to their administrative powers ; but many
so-called urban districts are more rural in character than
some so-called rural districts, which have not yet adopted
urban systems of drainage, etc. The towns are constantly
growing past old boundaries, and neighbouring villages
gradually assume urban characteristics, while still in the
midst of agricultural country.

5. The following table shows the growth of the population
of England and Wales, Scotland and Ireland, separately and
together. As an example of further analysis the populations
of London and of the principal manufactviring counties are
also shown. The county of London contains about 120 square
miles, the great part, but by no means all, of which is thickly
populated ; adjacent to it are considerable districts which for
many purposes should be included in London, and the popula-
tion of London and its dependent environment may be put
at about six and a half millions. In the table the northern
counties include the whole of Cheshire, Lancashire, Yorkshire
(West Riding), Durham and Northumberland ; the midland
counties include the counties of Derby, Leicester, Nottingham,
Northampton, Stafford, Warwick, Worcester, Monmouth and
Glamorgan. It will be noticed that the population of each
of the three selected regions has doubled in fifty years, while
that of the rest of England and Wales has increased only
50 per cent.

Growth of Population

United
Kingdom.

OOOO's

England

and
Wales.
OOOO's

Scotland
OOO's

Ireland.
OOO's

Present Mining and

County of Manufacturing

London. Counties.

Northern. Midland.
OOOO's

Rest of

England

and Wales.

1851

. 2737

1793

2889

6552

236

451

276

829

1861

. 2893

2007

3062

5799

281

529

321

876

1871

. 3148

2271

3360

5412

326

628

364

953

1881

. 3488

2597

3736

5175

383

757

427

1030

1891

. 3773

2900

4026

4705

422

862

489

1126

1901

. 4146

3253

4472

4458

454

976

568

1255

90 AN ELEMENTARY MANUAL OF STATISTICS

6. Closely connected with the distribution by locality is the
distribution by occupation. This classification is extremely
difficult, and it is prudent to make only those comparisons
which are given in the census volumes, and to regard even
them with suspicion, unless one has time to go into the
question in minute detail, reading the text of the Gene-
ral Report for each census, and studying the changes in
classification.

In using the following table it must be realized that the
figures are per 1,000 of the selected part of the population,
not absolute numbers, and that one such division can only
grow at the expense of another. For example, the actual
number of females working in connection with Textile Fabrics
was greater in 1901 than in 1881. Further, it must be
remembered that the groups are not homogeneous (see
p. 65 above) either in age or in occupation. The number
of occupied children tends to diminish as educational require-
ments are enforced ; this accounts, for example, for part of
the diminution under the heading " agriculture." The table
is greatly contracted, and only suggests broad outlines for
investigation.

Under the heading "Professional," etc., are included
those engaged in government, central or local, and their
subordinates, the army and navy on land or in port, and
members of the professions and their assistants. " Domestic "
includes indoor and outdoor servants and laundry-workers.
" Commercial " includes merchants, dealers, " travellers," and
clerks. " Transport " includes railways (but not railway con-
struction), roads, rivers, docks and the telegraph and tele-
phone services. [By the grotesqueness of the census tabu-
lation the Post Ofiice comes under heading I, 1. "National
Government."] " Metals " includes all work in metals, ex-
cept mining, and the manufacture of tools, machinery, and
engines. "Building" includes navvies and road labourers.
The other headings are clear. Sailors and soldiers are only
included in the census enumeration when on land or in port
at or within a few days of the date of the census.

THE POPULATION CENSUS

The residual heading in the census, " Without specifie d
occupations or unoccupied" has no relation whatever to
" unemployed " ; it includes among the males in England
and Wales 262,000 persons retired from business,* 26,000
pensioners,! 93,000 "living on their own means," and
1,600,000 others, "including students." The number of
women " without specified occupations " is of course very
much greater.

Groups of Occupations in the United Kingdom

Per 1,000 Males

Per 1,000 Females

over 10 years.

1881.

1891.

1901.

1881.

1891.

1901.

Professional, etc.

. 46

Domestic

. 8

142

132

122

Commercial

. 30

Transport

. 75

Agriculture

. 188

162

136

Mining .

—

Metals

. 75

Building .

. 74

—

Textiles ....

. 48

Dress ....

. 35

Food and Lodging .

. 54

Other Manufactures, etc. .

. 61

Undefined

. 85

Total occupied

. 827

827

834

335

330

316

Retired or unoccupied

. 173

173

166

665

670

684

1,000

Actual total number of per-

)

sons over 10 years (OOOO's

y 1255

1389

1554

1350

1580

1680

omitted)

7. Apart from the census we have other information as to
numbers of persons actually employed in textile factories,
in or about coal mines, and in some other occupations, from
returns made by employers to inspectors or to the Home
Office. The difference between the two sets of statistics
affords a good example of the necessity of examining

* Other than the Army, Navy, Church or Medicine; these are
tabulated under their professions.

•[• Before the existence of " old age pensions,"

92 AN ELEMENTARY MANUAL OF STATISTICS

definitions and methods of collection, and also of the
necessary approximate character of all general estimates.

Number of persons occupied in

Census 1901, United
Kingdom.

Home Office

Statistics.

Cotton manufactures ....

Woollen and worsted manufactures .

Silk manufactures ....

Hosiery manufactures

Lace manufactures

Other textile manufactures .

Coal mines ... . .

545,959
239,843
37,459
60,950
41,453
73,398
752,626

522,623

259,909

31,555

38,549

17,902

158,815

792,648

In the case of cotton and wool, practically all persons
occupied work in mills or factories, and the difference
between the two columns is to be partly accounted for by
the number of factories in which both cotton and wool are
used, and the inclusion or exclusion of supplementary pro-
cesses {e.g. dyeing). There are many persons working in
hosiery and lace outside factories ; hence the difference
shown in the numbers.

8. The areas of all the districts and sub-districts are stated
in the county reports, and the density of the population (the
numbers of persons per acre or per square mile) can be
worked out in minute detail. It is important for this
purpose to take sufficiently small areas, for the least con-
sideration shows that for nearly all practical purposes the
variation over the area of a square mile is more important
than that from county to county.

The table on page 93 shows an analysis for the neighbour-
hood of Middlesbrough, as compared with England as a whole
and in contrast to Shoreditch in particular.

In this case we have the crowded town of Middlesbrough,
occupying 4 square miles, separated on one side from the
municipal Borough of Thomaby, less crowded and occupying
3 square miles, by 17 square miles of rather empty agricultural
country, while on the other side of it lies an area of 10^ square
miles of urban district containing on the average about one

THE POPULATION CENSUS

house to an acre. This great diversity would be entirely con-
cealed if we had not analysed the registration district into its
parts. Maps, which are shaded to show the density of popula-

1901.

Area
(including

Number

inland water).
Acres.

Population.

of persons

per square

mile.

of acres

per
person.

England and Wales .

37,327,479

32,527,843

558

1-15

Yorkshire (North Riding)

(Registration County)

1,277,104

375,918

188

3-40

Middlesbrough —

Whole Registration District

22,254

139,773

4,022

■16

Contained Registration Sub-

distriots —

rl. Middlesbrough .

7,480

92,668

7,929

•08

J 2. Ormeshy

6,786

30,326

2,860

■22

(3. Thomaby .

7,988

16,779

1,344

•48

The same in detail —

'County Borough of

Middlesbrough .

2,685

91,302

21,760

•03

l.-{ Linthorpe t •

515

417

516

1^24

Martont

3,121

819

168

3-81

West Acklam -f

1,159

130

8^92

rBston Urban Districts .

2,453

11,199

2,922

•22

2.-1 Normanby „

1,500

9,645

4,115

•16

[Ormesby „

2,833

9,482

2,149

•30

■Hemlingtont

1,119

115

9-73

Ingleby Barwick t

1,519

124

12^25

Maltbyt •

1,117

139

8 00

Staintont •
Thomaby-on-Tees

2,306

347

6-65

(Municipal Borough)

1,927

16,054

5,332

•12

Rural District of Middles-

brough, containing

parishes marked f

10,856

2,091

123

5^19

London —

Shoreditch

1,131

206,180

116,530

•005

tion, are apt to be extremely misleading because the changes
of colour can hardly be made over sufficiently small areas, and
if they were, the great detail would be confusing.

94 AN ELEMENTARY MANUAL OF STATISTICS

9. The census affords a test of the amount of overcrowding,
detail being given for every town, urban district and rural
district. The test taken is the number of persons relatively
to the number of rooms occupied, and tenements in which
there are more than two persons to a room are considered to
be overcrowded. Of course, this test is very rough, for over-
crowding depends also on the size of the rooms and the age
and sex of the persons, to name only two considerations, but
it affords some means of measuring progress and of noticing
where serious overcrowding is prevalent.

The following table shows part of the information for
London, and for the aggregates of County Boroughs, of other
urban districts, and of rural districts in England and Wales.

OVBRCROWDINe IN ENGLAND AND WALBS, 1901
Number of tenements (00' s throughout)

County of

County

other

Rural

Tenements of Occupied by

London.

Boroughs.

Districts.

1 room. 1 or 2 persons

1087

409

280

More than 2 persons .

408

159

2 rooms. 4 or less persons .

1511

1421

1366

808

More than 4 persons .

503

402

381

190

3 rooms. 6 or less persons .

1576

1981

1808

1410

More than 6 persons .

240

326

273

187

4 rooms. 8 or less persons .

1298

4430

5388

4094

More than 8 persons .

209

277

173

5 rooms or more ....

3476

10230

14242

9556

Total number of tenements .

10196

19567

24098

16508

Number of overcrowded tenements

1248

1096

1014

568

Per cent, of tenements o-v ercrowded

12-2

5-6

4-2

3-4

Per cent, of population in over-

crowded tenements .

160

8-0

6-0

5-8

10. The census affords no complete means of distinguish-
ing industrial rank ; the director, manager, foreman, artizan
and labourer are not distinguished as such, for the main lines
of demarcation are between industry and industry, and the

THE POPULATION CENSUS 95

subordinate divisions are according to the general processes
(the spinning- as distinguished from the carding-room), and
not according to occupation (the spinner as distinguished
from the piecer). Nor does it show the social grading.
Mr. Booth {London: Final Volume, pp. 1-9) has made a
very ingenious social classification of the families of London,
according to the number of persons per room or rooms per
person for the families who do not employ servants, and
according to the number of persons per servant for those who
do. The information was specially extracted from the census
forms, and has not been obtained for the country as a whole.
11. The population in 1901 is of course equal to that of
1891, together with the number of births and immigrants
between the census dates, less the number of deaths and
emigrants. Unfortunately, the emigration statistics are not
sufficiently complete to allow the census to be kept continu-
ally up to date. We have rather to work backwards to find
the net result of migration.

Ungland and Wales. —

coo's

Population, 1891 . . ... 29,003

Births, 1891-01 .... 9,156

38,159
Deaths, 1891-01 5,563

Population in 1901 if no migration . . . 32,596
Enumerated population 32,528

Deduced excess of emigrants over immigrants . 68

The excess of the number of births over deaths is called
the " natural increase of population."

It is necessary for many purposes to estimate the popula-
tion at intermediate dates. The most accurate method would
probably be to make the best estimate possible from the
migration statistics, whose effect can be checked every ten
years, and combine these with the recorded numbers of
births and deaths.

96 AN ELEMENTARY MANUAL OP STATISTICS

Another way is to assume that the population increases
continually in geometric progression ; this rate was equal to
•95 % per annum for the United Kingdom between 1891 and
1901° and to "79 % between 1881 and 1891 ; it is clear that
some process of " smoothing " is necessary to pass from one
rate to the other in 1891.

The following table shows various methods of estimating
the numbers for the United Kingdom at the middle of each
year, 1891 to 1901.

Population in 1891

1901

Excess

coo's
37,802
41^1

3,749

Logarithms

. 4-57751
4-61859
10) - 04108
-004108

Arithmetic
progression.

OOO's

Geometric progression

Computed
from Births and

Deaths less
69,000 net emi-
grants annually.
OOO's

Official

Logarithms.

Numbers.
OOO's

estimate.
OOO's

1892

38,177

4-58162

38,162

38,140

38,134

1893

38,552

4-58573

38,524

38,494

38,490

1894

38,927

4-58983

38,889

38,881

38,858

1895

39,302

4-59394

39,260

39,265

39,220

1896

39,676

4-59805

39,633

39,655

39,598

1897

40,051

4-60216

40,009

40,063

39,986

1898

40,426

4-60627

40,390

40,453

40,379

1899

40,801

4-61037

40,773

40,827

40,772

1900

41,176

4-61448

41,161

41,181

41,152

The first column assumes equal annual increments of
3,749,*""' persons ; the second method assumes an annual rate
•95 % (log 1-0095 = -00411) ; for the third method the births
and deaths in the United Kingdom are taken from the Statis-
tical Abstract,* and it is calculated that net emigration must
account for a loss of 590,000 in the decade, including about
520,000 from Scotland and Ireland. It is not stated how the
ofiicial estimate is obtained.

The first method is the most rapid, and agrees with the
others within 2 per 1,000. The more involved method, com-

* Actually to obtain the number of births between mid-1891 and
mid-1892, half the total numbers for 1891 and 1892 together is taken.
Similarly for deaths.

THE POPULATION CENSUS 97

bining numbers of birtlis and deaths and emigrants, is likely
to be the most correct, if the migration figures are studied
more minutely. In the decade considered difficulties arise
from the absence of troops in South Africa.

Any of the above methods can, and one or other must, be
used for estimating the inhabitants of a county district or
town ; the " natural " increase is known from registration, but
here is grave risk of error due to migration. The difficulties
are accentuated when we estimate the population in (say) 1909,
before we have the census of 1911. We may, however, take
the "natural" increase, and compare it with the increase
that the previous intercensal rate of growth shows; we can
base another estimate on the number of school children ;
and in some cases check the result from the number of
houses rated, but this is difficult. If these four methods
agree, our estimate is good ; their disagreement is a measure
of the inaccuracy of the result. Local knowledge will some-
times allow the better of the four estimates to be cliosen.*

12. The previous paragraphs deal with only a few of the
very large number of problems and results of interest that
arise from the census volumes. In conclusion, we will deal
very briefly with the statistics of age. Age is stated inac-
curately for the very young (through misreading of the
instructions), for the very old (through ignorance or through
the desire to magnify old age), and by women who are un-
willing to confess even under the cover of secrecy to advancing
age, and generally there is a tendency to return the age at
the nearest round number, instead of at the last birthday ;
this would be corrected if the date of birth were stated.
There will in future no doubt be a tendency to overstate
age, with the idea that an old age pension may depend on
it.

To overcome the concentration at round numbers, ages are
tabulated as between 25-35, 35-45, etc. The other mistakes

* Students who wish to study the methods in use should consult
papers by Mr. Waters, p. 293, and by Mr. Hayward, p. 434 of the
Statistical Journal, 1901, and follow up the references there given.
H

98 AN ELEMENTARY MANUAL OP STATISTICS

cannot be completely rectified, but they can be checked by
two different methods. First, there is the record of persons
at the various ages at all the previous censuses, and the
registers of deaths according to age and of births ; from these
the number surviving can be estimated, and the differences
found must be attributed to migration * or mis-statement of
age; the tables (18 and 19 of General Report of Census)
show the existence of some of the mis-statements already
named, but in general confirm the accuracy of the answers.
Secondly, it is certain that in a large population the
numbers at successive ages must result in a continuous and
regular group ; there cannot be a great number at 30, and
relatively few at 29 and 31. The application of this principle
is the basis of the life table, the survival table, the tabulated
death-rates according to ages, and the other tables which
supply actuaries with material for their calculations. The
method of smoothing in the diagram, p. 37, depends on the
same idea. It is beyond our scope to discuss here the
mathematical methods which are employed. The Census
General Report gives the result of the " graduation " for the
whole population year by year from Oto 105 years (Tables 20
and 21). The following contracted table is important as show-
ing the relative number of young and old, and of the two sexes.

NuMBBH PER 10,000 Persons Enumerated in England and Wales

IN 1901

Under 5 years .

5 and under 15

15 „ 25

25
35
45
55
65
75
85 and over

45
55
65
75
85

Males.

Females

570

572

1,048

1,051

947

1,011

764

852

594

635

429

463

279

318

147

184

4,835 5,165

* Or temporary absence in the case of soldiers.

THE POPULATION CENSUS 99

Note on certain Divisions according to which the
population is tabulated in the Census of England and
Wales.

Ancient Counties are the old counties, 40 in England, 12 in
Wales, which have been only slightly changed in historical
times by the merging of their detached parts in the counties
by which these are surrounded ; iu the case of Worcester-
shire considerable parts are still detached.

London was constituted as a separate administrative
county, carved out of Middlesex, Kent and Surrey, in 1888.

Registration Counties are groups of registration districts,
covering to a great extent the same areas as the Ancient
Counties by whose names they are called. The registration
districts are simply the Poor Law Parishes and Unions utilized
for registration purposes, births, marriages and deaths, as well
as census enumeration and tabulation. In connection with
the Poor Law Reforms of 1834 parishes were grouped into
Unions for Poor Law purposes round convenient centres, and
county boundaries were generally ignored. Consequently the
groups of registration districts which form a registration county
overlap the ancient county boundaries seriously ; for example,
the populations of the Ancient and the Registration County
of Derbyshire were 620,000 and 490,000 respectively in 1901,*
but in most cases the differences are less considerable.

The registration districts are divided into suh-distrids, and
each sub-district is made up of one or more civil parishes.
The civil parish is the smallest unit for Poor Law adminis-
trative purposes, but is not used for registration.

The statistics relating to the registration counties are sum-
marized for some purposes in eleven Divisions, viz. : London,
South-Eastern, South Midland, Eastern, South-Western,
West Midland, North Midland, North- Western, Yorkshire,
Northern, and Welsh.

Administrative Counties. These date from the Local
Government Act of 1888, which established County Councils.
Several of the old counties were divided for this purpose into
* Summary Tables of the Census, Table II.

H 2

100 AN ELEMENTARY MANUAL OF STATISTICS

two or more administrative counties (e. g. The Parts of Hol-
land, of Kesteven and of Lindsey in Lincolnshire, East and
West Sussex), so that there are now 51 altogether in England
and, as before, 12 in Wales. Boroughs which contained
over 50,000 persons in 1881, and a few others which had
before enjoyed some independence, were left outside the
administrative counties and called County Bm'oughs; other
boroughs which have since 1881 successfully claimed the
possession of a population of 50,000 have been raised to the
same rank. There were 68 county boroughs in England
and Wales in 1901. Many minor adjustments of county
boundaries were made, but, except for the separation of
London, the administrative counties (wl^en the subdivision,
as in Sussex, is ignored), together with the county boroughs
they surround, are nearly co-extensive with the Ancient
Counties.

Each administrative county (except London) is divided
into Urban and Bural Districts. The urban districts are either
boroughs or simply urban districts.* Boroughs are cities or
towns which have been incorporated ; each has a city or town
council consisting of the Mayor, the Aldermen and the
Councillors, whereas each other urban district has an
urban district council with chairman and councillors. Most
independent towns of considerable size or of ancient origin
are incorporated. In the Census Reports boroughs are dis-
tinguished as C.B. (county borough) or M.B. (municipal
borough), but strictly the latter includes the former. Other
urban districts are those regions which have been constituted
as such, because of their density of population or of their urban
character, from time to time by the Local Government
Board; they have special powers of administration, chiefly
for sanitary and engineering purposes ; the most populous of

* The county boroughs are sometimes classified with, sometimes
apart from urban districts. Also they are sometimes included in and
sometimes excluded from administrative counties in summary statistics.
The County Borough of York stands partly in each of the three Ridings.
Great care is necessary in reading the headings of tables on these
accounts.

THE POPULATION CENSUS 101

them are on the growing outskirts of boroughs in which their
destiny is to be included, others are mining or scattered
manufacturing districts.

The boroughs and other urban districts having been sub-
tracted from the county, the remainder consists of rural
districts, each of which possesses a rural district council. Each
urban and each rural district consists of a civil parish or
group of civil parishes; the parishes in the rural districts
have some powers of self-government exercised through
the parish councils.

Civil parishes are thus grouped together in one way to make
urban and rural districts and in another to make registration
sub-districts. An urban district is in general part of a
registration sub-district; a rural district in general the
remainder of a registration district when the urban districts,
if any, are subtracted, the main exceptions being when
the registration district is divided by the boundary of an
administrative county.

London, for which the administrative and registration
counties coincide, is under special laws; it consists of the
City of London (with its Lord Mayor) and the City of
Westminster and 27 Metropolitan Bwoughs (each with a
Mayor).

For most practical purposes the administrative counties
and county boroughs have superseded the Ancient Counties.
Birthplaces, however, are supposed to be recorded for the
census according to the latter.

The boundaries of civil parishes have been adjusted for
this grouping into distritts. Ecclesiastical parishes may either
coincide with ancient or with new civil parishes, or they have
been formed by subdividing former parishes, or by carving
out a new parish when the population required it.

The division into parliamentary constituencies does not
necessarily coincide with any of the divisions already named.

CHAPTER II

VITAL STATISTICS *

1. The most easily accessible source of statistics of births,
marriages and deaths is the Registrar-General's Annual
Report.! The extracts from it in the Statistical Abstract are
insufficient for most purposes. The sources of the Registrar-
General's statistics are the familiar marriage and death certifi-
cates and register of births, filled in by those responsible on
these important occasions. The registration districts are the
same as those used in the'population census.

Birth- and death-rates are obtained by multiplying the
number of births and deaths recorded in a year in a district,
great or small, by 1,000 and dividing by the estimated
population of the district ; the resulting rates are generally
given to one place of decimals (thus : 15 "3 per 1,000), and
in the last chapter it was seen that the population of the
whole of England and Wales, at any rate, could be estimated
with sufficient accuracy. The marriage-rate is obtained by
multiplying the number of marriages by two to get the
number of persons and proceeding as before.

2. The following table shows how these birth- and death-
rates have fallen in recent years in England and Wales.
Similar phenomena are observed in most civilized countries.

* Readers who desire more than this very slight summary should
consult Vital Statistics, by Dr. Newsholme, Medical Officer of the Local
Government Board. See also Dr. Newsholme's and Dr. Dudfield's
papers in the Statistical Journal, 1905, 1906, and 1908, and Bertillon's
Gours 4Ume.ntaire de Statistigiie Administrative, Ch. VII, XIII, and
XXVI-XXXII.

•f There are also weekly and quarterly reports and an annual summary
for London and large towns ; and a Decennial Supplement (of which
the last was published in August 1908), givingj comparative statistics
and much detailed information in Part I, and the relation of deaths to
occupations in Part II. The reports of local Medical Officers of Health
for districts throughout the country may be consulted with advantage.

102

VITAL STATISTICS

103

England and Wales
Kates per 1,000 of the Population.

Births.

Deaths.

Marriages.

1871-5 Annual average

35-5

22-0

17-1

76-80 „

35-4

20-8

15-3

81-85

33-5

19-4

15-1

86-90

31-4

18-9

14-7

91-95

30-5

18-7

15-2

96-00 „ „

29-3

17-7

16-1

1901-05

28-1

16-0

15-6

1906

27-1

15-4

15-6

1907

26-3

15-0

15-8

1908

26-5

14-7

14-9

It is believed that births are adequately registered, but
the possibility should be borne in mind that the regulations
recently put in force for the immediate notification of a birth
may bring the registration more up to date and appear to
show an increased birth-rate in 1909.

3. If the constitution by age and sex, as shown on p. 98,
was the same for all countries and districts, or even constant
for the same place over a period of years, we could use the
rates here given for comparative purposes. Actually there
has been a considerable change in England and Wales in the
last thirty years ; indeed, the falls in the birth- and death-
rates are almost * certain to increase the average age of the
population.

Age Composition op the Population of England and Wales,
Pee 1,000.

1871. 1901.

Males.

Females.

Males.

Females.

Under 5 years

. 68

5-20 „

. 161

161

154

156

20-45 „

. 166

182

181

199

45-65 „

65- „

. 22

487 1 513 484 1 516

All between 20 and 45 per 1,000 348 380

* If the death-rate had been reduced only for infants, the two falls
might compensate each other.

t Here and in many other tables both items and totals are stated in
round numbers, and the sum of the items as given is not necessarily
equal to the total.

.104 AN ELEMENTARY MANUAL OF STATISTICS

It is clear that so great a change in the relative numbers
of the young, the old, and those in the prime of life, must
affect the birth- and marriage-rates if these are simply based
on the whole population. The problem is to get a true
average (see p. 68) in which the number of marriages is
referred to the number of marriageable people, and the
number of births to the number of married people (with
reference also to age *), and to standardize a means of com-
parison with the same purpose as that used for correcting the
death-rate discussed below. Eeaders interested in this ex-
tremely important problem should devote careful attention to
the articles in the Statistical Journal referred to on p. 102.

4f. The general death-rate is also greatly affected by such
differences, for the rates differ greatly according to age and sex.

Death-Eates at Various Ages, England and Wales

Average

1891-96.

Average 1901-05.

Ages.

Males.

Females.

Males.

Females.

0- 6

62-9

52-8

53-8

44-9

5-10

4-5

4-6

3-6

3-7

10-15

2-6

2-8

2-1

2-2

15-20

4-0

3-2

3-0

20-25

5-2

4-8

4-3

3-6

25-35

7-1

6-6

5-9

5-0

35-45

12-0

10-2

9-9

8-2

45-55

19-6

15-2

17-3

13-3

55-65

35-9

29-5

32-9

25-7

65-75

72-5

63-1

67-2

56-4

75-85

149-3

134-1

137-4

121-5

85-

291-6

264-2

283-0

261-3

A.11 ages

20-0

19-8

17-1

15-0

The rate is reckoned for each sex-age-group as the number
of deaths in a year per 1,000 of the estimated number living
in the group at the middle of the year. The death-rate is
greatest immediately after birth. Infant mortality is always
reckoned as the number of infants under one year old who

* See Seventieth Annual Bsport of the Registrar-Oeneral, p. xxv, for
comparison of the birth-rates for thirty years on two methods.

VITAL STATISTICS

105

die in a given year per 1,000 born alive in that year, and
differs from the death-rate as just defined. The following
table shows that this mortality is very high, but that it has
fallen in the most recent years.

Infant Mortality

England and Wales —

1871-80

1881-90

1891-95

1896-1900

1901

1902

1903

1904

1905

1906

1907

1908

149

142

151

156

151

133

132

145

128

132

118

121

A population containing an excess of persons between the
ages of 5 and 45, and few infants or old people, will show a
smaller general death-rate than a normal population if the
rates at all the different age groups are the same in both
populations.

5. The method of correcting or adjusting the death-rates
for comparison of two dissimilar populations is important,
and, though complicated in idea, is easy to apply.

The method generally used e.g. in the Annual Summary
of the Births, etc., in London and other Large Towns is as
follows : The death-rates for all the towns concerned are first
calculated on the hypothesis that in each town the death-
rates are equal to those of England and Wales (say in 1891-
1900), age group for age group; the differences so found
between the general rates from town to town are due solely
to the differences of relative numbers at various ages. The

Year 1901

nistricts or Towns.

standard
death-rate.

Correction
factor.

Recorded

death-rate

1901.

Corrected

death-rate

1901.

England and Wales .

19-15

16-90

33 large towns .

17-72

1-0806

18-59

20-09

Rest of England and

Wales .

19-47

•9835

15-99

15-73

London

17-97

1-0656

17-63

18-79

Plymouth .

19-70

•9720

17-89

17-39

Oldham .

16-72

1-1453

19-64

22-49

Manchester

16-90

1-1331

22-10

25-04

Notice that the correction changes the order of Plymouth relative to London.

106 AN ELEMENTARY MANUAL OF STATISTICS

resulting numbers are called the " standard death-rates " for
the towns. Thus, if London has fewer children and old people
than the country in general, its standard is low ; if Plymouth
has many children its standard is high. The standards given
in the table are based on the statistics of 1891-1900.

The quotient obtained by dividing the standard (19'15) for
England and Wales by the standard for the district is called
the correction factor, and is given in the table for six districts.
This is calculated only now and again, and then applied to
the figures that are recorded year by year as they come.

In the third column the recorded (or crude) death-rates
are given for some districts taken at random.

The recorded death-rate multiplied by the correction factor
is called the " corrected death-rate."

Example, — London compared with England and Wales.

Correction factor = „ .^ = 1"0656. Corrected death-rate

= (recorded death-rate) 17-6S x 1'0656 = 18'79. Hence we
see that corrected death-rate = recorded death-rate x

-X — 3 — T-i — I ■ If the town standard is low the process

standard for town ^

raises the recorded rate in the proportion that it should be

raised to counteract the effect of age and sex distribution,

as shown by the standard rates.

This is not the most obvious process by which a correction

could be made, but it proves to be the easiest to compute,

and it can be shown by the principles of averages to yield

approximately the same result as the following apparently

simpler method : Calculate the death-rates for each age group

in the town ; apply these to a standard population group by

group; the result is the corrected rate, freed from the accident

of age distribution. This method would, however, need the

estimation of the age distribution every time, and some

dozens of arithmetic processes for each town instead of one ; *

* If Sj, % . . . are the relative numbersin the age groups in England and
Wales, and t„ tg • ■ • in the groups in an assigned town in the standard
period, and if Dj, D2 . . . are the corresponding standard death-rates,

VITAL STATISTICS 107

it is used, nevertheless, in the Annual Reports and given for
all the registration counties.

6. The importance to public officials of the study of com-
parative death-rates can hardly be over-estimated. If the
death-rate in a district is above that in similar districts there
is a priori something wrong, and very careful analysis is
needed to determine what it is. Death-rates depend not
only on age and sex, whose effect can be tested as in the
previous paragraph, but on occupation, as to which statistics
are given once in ten years by the Registrar-General, and on
occupation combined with age; death-rates are, of course,
influenced also by epidemics and by catastrophes, and the
years affected in this way must be ruled out of comparison.
The most important subject for study at the present time is
infantile mortality, which may be regarded as of such a distinct
character from general mortality that the latter should be
restricted to the rate per persons over five years. There is
no doubt that a great part of infantile mortality can be
avoided ; in considering its magnitude attention should be
directed to the age (in weeks and months) of the infant, to
the economic position of the parents, to the cause of death,
with special reference to obviously avoidable causes and to
the annual epidemic of summer diarrhoea, and to the effect
on the rate of the presence in the district of workhouses,
hospitals and other institutions, where the presence of
specially feeble infants may in some cases be expected.

7. Problems relating to sickness and mortality naturally
come within the province of medical officers of health, and in
many districts these officers present admirable annual reports,

and di, d^ . . . the recorded death-rates for the town in any year, and

if 2s = 1000 = tt, then ^ , are the standard rates for

1000 1000

England and Wales and the town, -^ the correcting factor, the

recorded death-rate, and — - — x -^ the corrected rate.
1000 2tD

The other method would be to take as the corrected rate.

1000

108 AN ELEMENTARY MANUAL OF STATISTICS

tackling the questions of most importance in their localities
with statistical and professional skill. It will perhaps be
useful to indicate the application of the methods sketched
in Part I above to this class of problems.

The most important method is that of averages in the form
of rates. Besides death-rates, etc., we have the " morbidity-
rate " (or " attack -rate ") which is the number of cases of a
particular disease (multiplied by 1,000 or some other round
number) divided by the population, and the " case fatality "
rate, which is the number of deaths due to a disease divided
by the number of cases. Here, as with death- and birth-rates,
the denominator must be chosen carefully ; for the morbidity-
rate the persons should be grouped by ages, districts, etc., so
that the classes with different degrees of liability to the
particular disease shall be considered separately. For the
" case fatality " rate, great care must be taken to include all
the cases, and to be certain of the diagnosis. If differences
of treatment (hospital or home) or the efficacy of protection
(vaccination, isolation, etc.) are in question, there is always the
risk that the ages or economic conditions of the classes con-
sidered may differ, and the groups must be made similar
before comparison is attempted.

All through vital statistics there is great risk of inade-
quacy of, and even of mistakes in, definition. These arise (i)
from intrinsic difficulty of classification and incomplete
standardization of description ; (ii) from unconscious personal
bias of the practitioner ; (iii) from the presence of two diseases
together, or a disease and an accident ; (iv) from the desire
to avoid the statement of the existence of certain classes of
disease (e. g. alcoholism). The presence of any of these may
affect the apparent death-rate from any cause, and also the
morbidity and case-fatality rate.

In considering questions of cause and effect, liability of
various classes, and results of different treatments, the
essential thing is to get the exact difference to be considered
clearly stated, and then to proceed to analysis by tabulation.
If the headings of the table prove to be clear and distinct,

VITAL STATISTICS 109

and to follow the differences needed in the problem, the
table is good and relevant. Tabulation, when it is not analysis,
should either be omitted to save space if quite unimportant,
or relegated to an appendix if the data may be wanted at
some other time, or fitted into standardized tables in a
statistical section if they are needed for comparison. The
main line of argument or of information should not be
interrupted by tables which do not give definite answers to
definite questions.

Accuracy. — There is very great risk in most vital statistics
of spurious accuracy. In the practical question as to whether
one rate is greater than another, after the classes concerned
have been made similar there remains natural variation ;
if all known circumstances were the same, differences would
still be found. All records of births, deaths, marriages,
sickness, must be regarded as samples; the greater the
number of persons considered the more accurate the average
obtained from the sample. The only non-mathematical test
of this accuracy is the test of subdivision (see Chapter VII
above), that is, the finding the amount of agreement if smaller
groups are taken ; the mathematical tests are extremely
important, but should only be used when thoroughly compre-
hended, and are therefore not summarized here. A very great
number of differences that are remarked on, prove on mathe-
matical examination to be only the result of chance variation,
and to be no more remarkable than (say) the throwing of
double-six twice in succession. Here we can only recommend
extreme caution in drawing conclusions.*

As a simple and obvious rule, based on the elementary
ideas of accuracy (Chapter II, above), the rate should never
be reckoned to more digits than there are in the numerator
(number of cases, etc.).

Diagrams should be used sparingly and with reference to
the methods discussed in Chapter V above ; they are often

* There is much to be said for the establishment of a medical-mathe-
matical office, to which statistical data could be sent for examination,
just as bacteriological tests are made.

110 AN ELEMENTARY MANUAL OF STATISTICS

specially useful in traciug the course of an epidemic, and in
the relation of the seasons to the incidence of some diseases.
(See Studies in Statistics, Dr. Longstaff.)

8. It is often remarked, and has great theoretic and practical
interest, that averages arising from apparently quite fortuitous
causes are nearly unchanged from date to date. The death-
rate attributed to " varicose veins " in England and Wales
was between 2'1 and 3"7 per million persons living every
year from 1875 to 1894 ; similarly the annual rate for
" accident or negligence " was in the same period 703, 662,
632, 667, 602, 589, 608, 583, 592, 567, 549, 540, 558, 528,
528, 565, 574, 553, 576, 537, a series of small variation with
a downward trend. It is this partial constancy in the total
of events based on very large numbers which makes insurance
possible. In these instances the events are nearly independent
of each other ; as a contrast notice the death-rates when the
events are not independent, owing to infection, or to fashion
in diagnosis ; e. g. Influenza, 1875 to 1894 : 19, 8, 8, 8, 10, 7,
4, 3, 4, 3, 5, 3, 3, 3, 2, 157, 574, 534, 325, 220.

It is when we obtain approximate constancy or a trend with
small variation over a series of observations, as in the case of
the general birth-, death- and marriage-rates, and the distri-
bution by sex and by age, that we can apply statistical
methods for the elucidation of problems and the tracing of
cause and efifect.

CHAPTER III

TEADE AND TRANSPORT

1. The statistics of the External Trade of the United
Kingdom are published as follows : —

Early in every month a cheap unbound account is issued
stating the quantity and value of the exports and imports of
each commodity, showing the principal sources and destina-
tions for each, with figures totalled for the months of the
current year, and comparative statistics for the two previous
years. Home produce is separated from foreign and
colonial. Accounts of the movement of bullion and of shipping
are also included. The details in this monthly issue are
subject to correction.

A bulky annual report is issued volume by volume. For
1908, the first volume was issued in June 1909 and contained
nearly the same headings as the monthly issues, without
shipping, and with comparisons for five years ; the second
volume followed in August and summarized the trade with
each country, giving details for each commodity. A supple-
mentary volume is issued later in the year, distinguishiug
the apparent and real destinations and sources of trade (see
p. 122 below). A separate report is issued for shipping
(in August in 1909).

The Statistical Abstract for the United Kingdom, issued
in August, summarizes all the statistics of trade and gives
complete detail as to commodities, but does not show
commodities in relation to countries, for which volume ii of
the Annual Report and the Supplement are the only sources.

2. The basis of these returns is as follows : The exporter
of goods or his agent is bound to send a statement of the

112 AN ELEMENTARY MANUAL OF STATISTICS

quantity and value of the goods he is exporting to the proper
customs officer, who in general accepts the statement ; but
every bale, etc., on board ship has to be accounted for
before the ship is " cleared," i. e. permitted to leave the
port.

All imports have to be passed through a custom-house ;
the importer or his agent hands a statement of the goods he
desires to have passed, and the customs officers examine the
goods with sufficient care to assess duty, if any, or to verify
the absence of dutiable goods. These officials fill in the
value, from current price lists or otherwise, if the importer
has made no statement of value. Returns of the values of
imports and exports are further checked at the Central
Customs Statistical Office, and inquiry is made if the entries
appear unusual or are incomplete.

In this process there is a good deal of room for inaccuracy
in detail, which may be important for special classes of goods ;
but there seems no reason to doubt that the descriptions and
quantities are stated on the whole with fair accuracy. The
values are often a matter of estimate (e. g. in the case of goods
exported for sale by a foreign agent), and our only security
for accuracy is that in a composite total (see p. 29 above)
errors which are not biassed tend to neutralize one another,
and that, though there are inducements in some cases to
exaggerate value, there are inducements in other cases to
under-value.

In the case of exports the value is intended to be that of
the goods after all internal transport and dock expenses are
paid, that is the value at which the goods are delivered free-
on-board (f. o. b.). For imports the value is intended to be
that of the goods before they are landed, and includes their
cost, insurance and freight (c. i. f.). Thus exports are valued
at the moment they pass out of the hands of British shore-
labour, and imports before they are handled or pay duty. If
the exchange were simply across a land frontier, and the goods
of one community were exchanged as a whole against the

TRADE AND TRANSPORT 113

goods of another, it is clear that the method described would
give equal values for imports and exports.

As a matter of fact goods are often quoted at prices to
include delivery ; in these cases the value has to be corrected
for the trade statistics.

3. The following table gives the total trade statistics for
1907.

United Kingdom

£ £

OOO's. OOO's.

Imports of Merchan- Exports of Produce of
dise . . . 645,808 A the United King-
Imports of Bullion . 73,072 dom . . . 426,035 B
Exports of Foreign and

Total Imports £718,880 Colonial Produce . 91,942

Total Exports of Mer-
chandise . . 517,977 C
Exports of Bullion . 67,787

Total Exports £585,764
Transhipments under bond £18,824 G.

The total A is always quoted as the value of imports, and
B is generally quoted as that of exports.

Goods landed may be transhipped either at the same or
another port under bond, that is, without passing out of the
control of the customs officials, in which case they are entered
as " Transhipment " (G) and not included in imports and
exports. Goods which pass out of control of the customs are
either for use or consumption in the United Kingdom, or for
sale again in another country ; all such are counted as im-
ports, but when imported goods come to be re-exported they
are declared as of foreign or colonial origin. The value A
of imports is then thus composed —

Merchandise only

OOO's.

Imports for consumption . . . £553,866 D
Imports for re-exportation . . . 91,942 E

Total £645,808

114 AN ELEMENTARY MANUAL OF STATISTICS

Since the goods are valued afresh for exportation, they are
presumably increased in value by the expense of handling
them in the country, and the value E is thus a little too
great.

" A" should be compared with C, and D with B.

Actually no theoretic line can be drawn between goods
which are (i) simply transhipped, (ii) goods which are re-
exported unchanged, (iii) goods which are done up with new
labels and re-exported, (iv) goods which form some constituent
part of a machine which is exported, (v) yarn which is ex-
ported when woven, (vi) wool which is spun and woven and
then exported, (i) is included in neither exports or imports,
(ii) is included in imports and in exports of foreign produce,
(iii) to (vi) are included in imports and in exports of produce
of the United Kingdom. It is not possible to correct this
method, but it is important to understand it and consider it
in the light of pp. 64-5 above.

Notice that bullion and specie, that is metallic and coined
gold and silver, enter nearly equally on both sides of the
account.

No special record is kept of trade from one port to another
of the United Kingdom or of islands in the British Seas
(Channel Islands, etc.).

4. Other countries have different methods of definition,
valuation and classification.* Before using their statistics, it
must be ascertained how imports for consumption, for re-
exportation with or without alteration, and exports of national
and foreign produce are treated, whether bullion and specie
are included, exactly what districts are included in the country
concerned, and whether there are any peculiarities in the
method of valuation.

The general method is as follows : Goods are valued with
the intention of producing results on the basis described
above for the United Kingdom. Bullion and specie are

* See Keports of the Committee of the British Association on " The
Accuracy and Comparability of British and Foreign Statistics of
International Trade," 1904 and 1905.

TRADE AND TRANSPORT 115

excluded. All goods entering and leaving the country are
included in totals of Qe.neral Imports and Exports ; goods for
consumption or use in the country and exports of goods which
have been produced or undergone any process of manufacture
in the country are included in totals of Special Imports and
Exports. General exports are thus greater than special
exports by the value of goods passed in and out of or
through the country, and similarly with imports; the
differences for exports and for imports are approximately
equal.

For the United Kingdom we should have

OOO's
General exports . . C + G £536,801
Special exports . . B £426,035

General imports . . A + G £664,632
Special imports (approx.) D £553,866

It should be noted that the United States value imports,
not on arrival at the port of destination, as is general, but at
the place of manufacture.

5. If we regard the international trade of the world as a
whole, a consignment forming part of the special exports of
one country may appear under general imports and exports
of all the countries it passes through, but will finish as a part
of the special imports of some one country. The same con-
signment will be worth more as imports than it was as exports
by the cost of transport (including freight, insurance,
transhipment and commissions). The following table shows
the relation of the special imports and exports of the
principal trading countries of the world for the year 1904.
The numbers given are subject to many minute corrections.
It is thus seen that imports on the whole are worth about
9% more than exports as a whole. The difference, rather
over £200,000,000, is received by those engaged in any
capacity in international transport, and of this a very large
share appertains to the citizens of the United Kingdom.

I 2

116 AN ELEMENTARY MANUAL OF STATISTICS

International Trade. [Merchandise only,

except when noted.]

1904.

Special

Excess of

Imports.

Exports.

Imports.

Exports.

OOOOOO's.

Eussia and Siberia

£69

£106

—

Finland

—

Norway

—

Sweden

—

Denmark .

—

Germany .

318

261

—

HoUand

200

165

—

Belgium

111

—

France

180

178

—

Switzerland *

—

Portugal

—

Spain .

—

Italy .

—

Austria-Hungary

—

Greece *

—

Bulgaria t •

—

Servia .

—

Roumania t

—

Total Continent o

Europe and Siberia

1,252

1,114

138

—

United Kingdom

481t

301

180

—

British India

110

—

Australia .

—

British South Africa

—

Canada

—

Other Colonies, etc.

—

Total British Kmpire

752

574

178

—

U.S.A.

205

299

Mexico t

Costa Rica f

Brazil f

—

Peru

—

Chile *

Uruguay

—

Argentina * .

—

Total America, othe

than Canada .

319

470

—

151

China .

Japan .

6
186

Grand total

2,411

2,225

Add 9% on U.S.A

Imports § Tote

i 2,430

2,225

205

—

* Including bullion and specie.

t In these countries general imports and exports are given, but the difference between
general and special must be small. J Imports less re-exports.

§ In order to make their value comparable with those of other countries which value
imports on arrival, while U.S.A. values them at place of manufacture. 9 is the value
of K given by the equation 2225(l+-i) = 2411+?2l2.

TRADE AND TRANSPORT 117

6. Imports are either received as capital loans, or as pay-
ment of interest, or in payment of shipping services, or in
payment of exports, or ia return for bullion and specie.
Except in the case of the gold-producing countries the balance
of bullion and specie is in general small. When a country
has lent capital (in the form of goods not immediately paid
for), interest will return to her year by year in the form of
goods. Imports are increased relatively to exports when
interest is received or capital called in ; exports are increased
relatively to imports when capital is sent out or interest is
paid. Put otherwise, an excess of imports is due to the earn-
ings of shipping together with the excess of interest over new
outputs of capital.* Interest or capital which appears as
exports for one country appears also as imports for another.

No such balance can be struck between the trade of any
two countries ; for example, goods sold by Australia to
Germany may be paid for by goods sold by Germany to
England, and other goods sold by England to Australia, to
take a simple instance ; it is only for the totality of external
trade that a balance is even conceivable.

7. The table on page 118 shows in some detail the values of
imports and exports since 1855 ; the statistics of imports
prior to 1855 were not computed on the same basis. To
follow the history of the external trade as a whole, smoothed
diagrams (the averages being taken over eight or more years),
should be constructed as on pp. 40-2 above. It will then be
seen that there has been a general but not uniform upward
trend throughout the period, concealed or accentuated by
considerable fluctuations.

8. The fluctuations both of imports aud of exports are
principally due to movements of price, and unless we eliminate
these we obtain a very imperfect view of the course of trade.
The following chapter shows how considerable these move-
ments have been. The method generally used for studying
the quantity, or volume, of trade, as distinguished from its

* For the statistics of interest and capital for the United Kingdom
see Mr. Paish's paper. Statistical Journal, 1909, pp. 465 sqq.

118 AN ELEMENTARY MANUAL OF STATISTICS

External Trade op the United Kingdom (OOOOOO's)

Imports, less re-exports.

Exports of Home Produce.

Declared value.

Estimated value

at prices of

1902.

Declared value.

Estimated value
at prices of 1902.

1870

£259

£160

£200

£142

1871

271

166

223

150

1S72

296

186

256

162

187S

315

195

255

160

1874

312

200

240

160

1875

316

20O

223

169

1876

319

223

201

155

1877

341

230

199

159

1878

316

231

193

160

1879

306

234

191

171

1880

348

264

223

194

1881

334

244

234

211

1882

348

268

241

211

1883

861

276

240

217

1884

327

268

233

220

1885

313

272

213

211

1686

294

270

213

222

188T

303

283

222

231

1888

323

294

234

242

1889

361

326

249

251

1890

356

324

263

260

1891

373

339

247

235

1892

359

339

227

1893

346

336

218

222

1894

360

364

216

230

1895

357

384

226

249

1896

385

410

240

261

1897

390

415

234

257

1898

410

437

233

266

1899

420

437

264 (265)»

272

1900

460

442

291 (284)
280 (271)

262

1901

454

267

1902

463

283 (278)

283

1903

473

469

291 (286)

291

1904

481

477

301 (296)

301

1906

487

473

330 (324)

827

1906

523

489

376 (867)
426 (416)

377 (866)

358

1907

554

600

880

1908

613

484

862

Averages.

1855-1858

£145

£117

1859-1863

184

132

1S64-1868

232

176

1869-1873

278

225

1874-1878

321

£217

211

£169

1879-1883

339

253

226

201

1884-1888

312

277

223

226

1889-1893

369

333

241

237

1894-1898

378

402

230

251

1899-1908

454

453

282 (276)«

276

1904-1908

612

486

362 (354)

344

* The numbers in braclcets exclude the value of ships built at home and sold to
foreigners. Before 1899 their values, which correspond, of course, to exports properly
deBned, were not ascertained or included. There is therefore a brealt of continuity

TRADE AND TRANSPORT 119

value, is as follows: The prices of all goods for which
definite quotations can be made are ascertained for a particular
year or short period; the quantities of goods exported or
imported are then valued in each separate year at these
standard prices; it is then assumed that the differences in
value shown for the goods which can be priced are typical for
all goods. E. g. to take an imaginary example —

Value of imports in (say) 1890, as stated in the accounts,
i. e. at the prices of 1890, £356 (millions). Take 1902 as
year of standard price. Suppose that £300 worth of the 1890
imports can be separately valued, and are found to be worth
£273 at 1902 prices, the prices in 1890 being higher than
those in 1902 ; then it is assumed that the whole £356 would
be reduced in the same ratio, viz. to £356 X |^ = £324, if
all could have been valued.

Such a calculation has been carried out year by year by
the ^cowomis^ newspaper, goods each year being valued at the
prices of the year before. Recently the Board of Trade has
issued annually a statement of the values each year since
190D, as they would have been if the 1900 prices had remained
unchanged. Prices, especially of coal and steel, were unusually
high in 1900, and 1900 is an unfortunate year to have chosen
as a standai'd ; exceptional years should always be avoided,
as likely to lead to erroneous conclusions. The table above
shows the values when 1902 is taken for the standard year.

The method is open to a good deal of criticism in detail,
but there is no doubt that it leads to results that are
substantially correct, at any rate over short periods.

It is interesting to notice how small are the actual fluctua-
tions in quantity, as indicated by the values at unchanged
prices, especially in imports. Consumption of goods and, to
a very great extent, production goes on with little change in
times of commercial inflation and depression.

The necessity of some such examination is emphasized by
the consideration that the rise of Id. per lb. in the price of
raw cotton raises the value of imports by about £8,000,000,
and since about four-fifths of the cotton manufacture is for

120 AN ELEMENTARY MANUAL OF STATISTICS

export, the value of exports is also raised by over £6,000,000
these immense changes would take place without any change
in quantity or in the work done by British capital and
labour.

9. The tabulation of the statistics of the foreign trade has
recently been greatly improved, and the new method has been
carried back to 1891 in the Statistical Abstract for 1905.*

The table on page 121 shows the summary statistics for the
new tabulation.

The complete meaning of the classification can only be seen
by looking at the detailed list in the Statistical Abstract ;
but it may be mentioned that commodities such as yarn and
pig-iron, which are the finished product of one process and the
raw material of another, are classed as "mainly manufactured."

In the lower part of the table is shown the values for coal,
the principal exported raw material, and of the principal
groups of manufactures.

In using the table it must be remembered that 1900 and
1907 were years of exceptionally high prices.

10. The original sources of imports and ultimate destina-
tions of exports cannot always be known. If, for example, wool
grown in Turkey were spun in Austria, woven in Germany,
sent by rail through Holland, manufactured into ready-made
clothes in Leeds, and sold in Canada, it would figure in the
export and import statistics of many countries, and its value
would be due to the co-operation of many nations. Again, if
goods are sent from London to Antwerp for sale, they may
pass on to Germany, Russia, Austria or Switzerland without
the English manufacturer knowing their destination. Till
recently imports have been only stated as from the country
from which they were last shipped, and exports have been
stated as to the country to which they were shipped direct.f
Thus Switzerland, Bolivia and Rhodesia, which have no sea-

* A more detailed examination of classes of manufactured goods is
given in Cd. 2337, Mem. xii, and continued in Od. 4954, pp. 48 sqq.

t It seems probable that exports have often in fact been entered as
going to the country to which they are sent on a through bill of lading,
whether directly by ship or not.

TRADE AND TRANSPORT

121

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122 AN ELEMENTARY MANUAL OF STATISTICS

board, have had no place in our statistics. German and
Russian goods have been entered as imported from Holland,
Austrian and Swiss goods from Belgium, and so on. From
1904 a second method has also been used, and the results
shown in a supplemental volume of the Annual Report and
in the Statistical Abstract.* Importers have stated the
country from which goods are actually consigned to them;
this is generally also the country in which they were
produced or manufactured or received their last process of
manufacture ; exporters have also stated the countries to
which goods were consigned, which are in general the ultimate
destination. The following short table shows the results for
certain European countries. The imports in the second
column are in a very different proportion from that in the first ;
the exports generally do not differ in the third significant figure.

Imports 1907.

Exports 1907

(including Foreign and

Colonial produce).

Received direct from

Consigned
from

Consignments

retained for

consumption.

Exported to

Consigned to

OOOOO's
Eussia . £314
Germany . 388
Holland . 368
Belgium . 283
Trance . 528
Austria . 11
Switzerland

329
572
160
175
463
68
84

1851

306
541
154
168
398
64
72

1703

191
567
190
134
335
54

1531

191
567
190
169
332
54
29

£1892

1532

A committee of the Board of Trade reported in 1908 in
favour of making the principal classifications according to
the countries of consignment in the future, and of dropping
after a few years that according to the countries of shipment.f

11. Shipping statistics call for little comment except as to
the meaning of tonnage (see note at the end of the chapter).

* The Abstract for 1907 shows the results as in the table here given.
That for 1908 gives the countries of shipment in Table 34, and of
consignments in Tables 35, 36.

t Minutes, Cd.4346; Report, Cd. 4345.

TRADE AND TRANSPORT 123

The Statistical Abstract gives a series of useful and easily
intelligible tables on the subject. Every ship is registered
as of a definite nationality, which is generally * that of her
owners. On entering a port of the United Kingdom the ship's
papers must be shown, stating whence she came, where she
last broke bulk, what cargo she carries, and her registered ton-
nage. She cannot leave the port till her papers have again been
seen, her next destination stated, and the necessary declara-
tions of her cargo are in order. In theory, nothing enters
or leaves the United Kingdom without official knowledge.

Steam- and sailing-ships are distinguished. It must be
remembered that a steamship carries much more cargo
between the same two countries in a year than a sailing-ship
of the same carrying power, owing to her greater speed and
more frequent journeys. Coastwise and foreign voyages are
distinguished ; coastwise means between any two ports in the
United Kingdom or islands in the British seas; foreign is
from or to a port in the United Kingdom to or from a port
in a foreign country or one of the British possessions.

None of the tables show the aggregate of the voyages of
British ships or any other measure of the work done by them,
and it must be realized that a great part of the merchant
navy carries cargo between distant ports without calling at
home at all. The share of our merchant navy in the
£200,000,000 shown on p. 116 cannot be estimated accurately.

The Abstract for Foreign Countries shows the aggregate
tonnage of the ships engaged in foreign trade under the flag
of the principal maritime nations in 1904.

Tons OOO's

Merchant Navies of

Sailing.

steam. Merchant Navies of

Sailing.

steam.

United Kingdom

. 1,803

8,752

Norway

. 809

643

Germany . . •

. 579

1,774

U.S.A.t .

. 345

554

Japan . .

. 327

797

Italy .

570

462

Spain . .

717

Sweden. . . .

266

408

France . .

. 653

696

* It is said that the United States have considerable holdings in ships

in the Atlantic trade registered as British.

t In addition U.S.A. had 2,351,000 tons sailing and 3,041,000 tons steam employed in
inland and coast navigation.

124 AN ELEMENTARY MANUAL OF STATISTICS

The measurement of tonnage is not quite uniform, and the
line excluding small vessels (whose aggregate tonnage is also
small) is drawn differently from country to country.

12. Railway statistics are deficient in the extreme for the
United Kingdom. In the Statistical Abstract the principal
known facts are summarized. Apart from finance, these are
the length of line open (distinguishing single from double or
more), the number of passengers, and the weight of minerals
and of other merchandise carried. There is no information
as to the average or aggregate distance travelled (passenger-
miles, and ton-miles), which is used in other countries to
measure the work done by railways. The totals are so crude
and heterogeneous as to be practically valueless (see p. 65
above), except that over a very few years the total weight
carried gives some indication of the upward and downward
movements of trade. In the official annual return on
railway working {e.g. Cd. 4804 for 1908) we have also the
totals of " train-miles,'' distinguishing passenger from goods,
that is, the aggregate of the number of miles run by all the
trains in a year. This regards as similar a mile run by a
crowded suburban train, by a long-distance express, and by
a nearly empty local train, and the total is evidently not of
general use.

Note. — Shipping tonnage. The definition and measure-
ment of tonnage are extremely complicated, as may be seen
from the Report on the Merchant Shipping Bill (H. of C.
256, 1907), where many examples are given. There are at
least four measurements of a ship's size or capacity : dis-
placement, burden, gross tonnage, and net or registered
tonnage. The displacement is the weight of the ship
(unloaded), which equals the weight of the water displaced ;
the burden represents its carrying capacity ; neither of these
are used in the general shipping statistics. The gross tonnage
is the number of times 100 cubic feet is contained in the
ship, measured according to certain rules; 100 cubic feet is
taken as representing the space occupied by a ton of cargo,
but a ton of coal occupies only about 45 cubic feet, and a ton

TRADE AND TRANSPORT 125

of water about 35 cubic feet ; light or loosely packed cargoes
occupy more. Net tonnage is obtained from gross by subtract-
ing according to artificial rules space occupied by the engines
(with an allowance for bunker and air space), by the crew's
quarters and the parts necessary for navigation ; the remain-
der (reckoning as before 100 cubic feet to the ton) is
supposed to represent the carrying capacity of the ship, and
is the registered tonnage. The rules for measurement and
deduction differ for diSerent nations, but there has been a
wide-spread movement in the direction of adopting the
British system. The British rules have been modified from
time to time. Actually the registered tonnage does not bear
any close relation to carrying capacity, and is extremely
artificial. All the recent shipping statistics are given in
registered (or net) tonnage ; formerly they were given in
gross tonnage, but the present method runs back far enough
for all practical purposes, and it is easy in comparative
statistics to see if there has been a change in this respect.
Sailing-ships' tonnage (which has, of course, no allowance for
propelling machinery) should be kept distinct from stearn-
siiips. Marine architects continually try to build so that the
registered tonnage shall be as low as possible, since dock dues
are charged in proportion to this tonnage, and they take
advantage of the rules of measurement so that the deductions
allowed siiall be as great as possible ; in other words, they try
to reduce the registered tonnage relatively to the carrying
capacity. It follows that the growth of shipping tonnage
shown in the tables tends to fall short of the real growth of
carrying capacity. Further, the Plimsoll mark, which regulates
the weight a ship can carry, has quite recently been raised
to allow a greater weight without altering the registered
tonnage. The general result is that the shipping statistics
cannot be used for any fine measurements, and are not
comparable over a long series of years.

CHAPTER IV

PRICES

1. Prices from the ordinary commercial standpoint are of
course to be found in the trade journals, and summaries from
time to time in the Economist and the Statist. From the statis-
tical point of view we are only concerned with the change in
particular prices over a series of years, and with general price
movements. For both purposes the most accessible informa-
tion is to be found in the tables of the Statistical Abstract,
which show the prices of exports, imports, cereals and
minerals, and in Mr. Sauerbeck's studies of price movements
published annually since 1886 in the March number of the
Statistical Journal. The Board of Trade Report on Whole-
sale and Retail Prices (H. of C. 321, 1903) contains a great
many records of prices over a long series of years, and
interesting charts showing the prices of wheat and of bread
since 1800 were published in the Labour Gazette, May 1909.

2. The great difficulty in the measurement of prices is in the
definition of the commodity to be measured. In the case of
the staple raw materials of manufacture, cotton, wool, iron,
etc., and the principal raw foods, wheat, sugar, etc., the
various grades are to a great extent standardized, and it is
only after the lapse of a considerable time that difficulties in
exact comparisons are felt ; for example, wheat prices can be
properly compared over (say) 20 years, but in a century the
kind of wheat commonly in use has changed immensely. As
the raw materials pass through the various stages of manu-
facture endless varieties are introduced and the goods con-
tinually change their character without changing their name,
or qualities which were commonly used fell out of fashion ;

126

PRICES 127

for these reasons it is not possible to measure the price of
such commodities as cotton yarn or cotton piece-goods over a
long period ; still less can we reckon the change in price of
ready-made clothes, of machinery, of bicycles, etc. Similarly
the change in character of live stock, of timber, of everything
of which the source varies or which can be modified by man,
is readily perceptible after even a few years.

3. The measurement of retail prices is so difficult that
neither government departments nor statisticians have as yet
made much progress with it. All the varieties of production
and all the changes of fashion have their full influence here.
It is seldom that goods can be exactly matched, even in
external appearance, after a few years ; and a more subtle
difficulty is present, for the actual quality of goods is very
frequently changed with no corresponding change in price,
customers demanding articles at the price they are used to,
and the manufacturer making slight changes in the constitu-
ents to preserve his profit. As an illustration of another
difficulty, it may be observed that the price of travelling one
mile by railway has been nominally Id since railways existed,
but the kind of accommodation and the speed of travelling
have changed completely, and a considerable proportion of the
third-class journeys made are in fact charged at a lower rate.
We therefore leave the whole problem of retail prices on one
side as too complex for the beginner.

4. The Board of Trade prices of imports and exports,
published in the Statistical Abstract, are obtained by rejecting
those commodities, such as pictures, horses, machinery,
miscellanea, etc., for which an average price is clearly an absur-
dity, by dividing the total value of imports or of exports for
the year by the total number of units of quantity for each
commodity not rejected. The price stated is thus always an
average, not a market quotation, and in some cases (e.g.
carpets and druggets) the divisor is not homogeneous. An
apparent change of price is often due to an actual change of
quality, and in some cases this change is cumulative, not
accidental ; for example, if the general run of " heavy broad

128 AN ELEMENTARY MANUAL OF STATISTICS

■s .

Index

2
1

Numbers.

3^,2

III

2ii 3

per

S 'A

cwt.

lb.

cwt.

lb.

cwt.

ton.

toil.

oz.

£.

1853-9

12-8

38-2 '16-4

306

17-3

33-8

3-4

9-3

61-6

155

122

142

1860-4

11-2

34-3

18-4

6-4

16-8

19-6

2-8

9-0

61-3

176

— .

146

1865-9

11-8

31-7

190

5-7

15-8

17-6

2-9

9-8

60-7

167

—

144

1870 .

10-5

32-1

17-2

4-47

14-4

19-6

2-96

9-5

60-5

162

138

1871 .

11'8

361

16-4

3-52

13-3

21-6

305

9-6

60-6

162

144

1872 .

12-4

36-3

16-8

4-24

14-5

19-6

5-04

15-5

60-2

160

157

1873 .

130

33-8 i 16-7

4-01

14-7

15-6

6-23

20-5

59-2

160

1874 .

12-1

30-7

17-0

3-62

14-7

16-6

4-73

17-0

58-3

154

150

147

1875 .

10-6

30-3

16-7

3-47

15-4

151

3-64

131

56-7

150

140

138

1876

10-4

29-4

16-4

3 02

14-5

14-7

3-12

10-8

52-7

143

130

137

1877 .

12-5

33-8

160

2-93

14-4

161

2-87

10-0

54-9

148

125

135

1878 .

110

29-3

15-3

2-80

13-9

15-3

2-67

9-3

526

137

120

125

1879 .

10-6

27-4

14-7

2 76

13-6

13-7

2-57

8-6
8-8

51-2

131

112

120

1880 .

11-1

29-2

13-5

2-94

13-7

17-3

3-20

52-2

137

115

127

1881 .

11-0

28-9

12-8

2-92

13-9

16-1

2-77

8-8

61-7

137

122

1882 .

10-7

28-7

12-6

2-93

12-3

14-5

2-82

51-6

135

114

121

1883 .

9-8

27-2

12-5

2-91

12-1

12-3

2-61

9-2

50'6

131

111

118

1884 .

8'4

20-9

11-8

2-85

12-1

14-1

2-32

9-2

50-7

122

106

110

1885 .

7-8

18-1

12-1

2-86

10-0

11-3

2-18

8-8

48-6

115

101

104

1886 .

7-5

16-7

11-8

2-49

9-1

11-2

2-16

8-3

45-4

109

100

1887 .

7-6

15-6

10-6

2-51

10-1

11-2

2-36

8-2

44-6

107

1888 .

7-7

17-5

11-0

2-59

9-8

12-4

2-13

8-3

42 '9

110

101

1889 .

7-7

19-7

10-8

2-64

9-8

14-2

2-51

i6-o

42-7

111

104

1890 .

7-8

16-3

10-6

2-67

10-3

13-3

305

12-4

47-7

110

106

104

1891 .

8-9

16-5

10-7

2 '59

9-4

12-2

2-63

12-0

45-1

110

105

104

1892 .

7-7

17-1

10-1

2-39

8-7

15-2

2-57

10-9

39-8

106

100

1893 .

6-4

18-4

9-7

2-43

8-7

13-2

2-35

9-8

35-6

103

1894 .

5-3

15-5

9-6

2-06

8-5

13-6

2-30

10-4

28-9

1895 .

5-5

13-3

9-6

1-94

8-1

11-1

2-40

9-2

29-9

1896 .

6-2

13-6

9-5

2-31

8-4

12-2

2-39

8-7

30-7

1897 .

7-4

12-3

9-4

2-09

8-0

11'7

2-41

8-8

27-6

1898 .

8-0

12-3

1-80

8-2

10-5

2-63

9-8

26-9

1899 .

6-7

12-6

8-8

1-91

8-6

12-7

3-47

10-5

27-4

1900 •

6-8

12-8

8-5

2-61

9-5

14-7

4-20

16-5

28-2

104

111

108

1901 .

6-6

12-2

7-7

2-57

7-5

13-5

313

13-7

27-2

100

105

101

1902 .

6-7

10-6

7-2

2-54

7-5

12-8

3-24

12'2

24-1

100

1903 .

6-8

10-7

7-7

2-80

8-3! 13-5

315

11-6

24-7

101

100

1904 .

7-0

12-3

7-2

313

8-7

13-7

2-92

11-0 26-4

101

100

101

1905 .

7-2

14-8

7-2

2-65

9-3

17-0

3-15

10-5 27-8

103

101

104

1906 .

7-0

11-6

7-4

3-09

10-2

22-6

3-50

10-8 30-9

107

105

111

1907 .

7-7

12-0

8-1

3-30

10-3

22-4

3-71

12-6 30-2

111

112

115

1908 .

8-4

13-0

3-03

9-3

16-6

3-17

12'6,24-4

106

107

105

Columns 1-6 are the " Average Import Prices " obtained by dividing the values of the
commodities imported, as stated in the Statistical Abstract, by the quantities there
stated. Columns 7 and 8 are " Export Prices " obtained in a similar way.

PRICES 129

woollen tissues, all ■wool," increased in breadth, perfection of
manufacture, and finish, it would still be entered under the
same category. The table on p. 128 gives examples of prices
(wheat, jute, and pig-iron) where the quality has probably
not changed much, of others (cotton and coal) where the
relative proportions of different qualities have probably changed
perceptibly, but not when few years only are considered,* and of
others (tea and sugar) where there has been almost a revolu-
tion in the trades. Evidently these prices need interpretation
by persons conversant with the industries with which they
are connected. Silver, on the other hand, is perfectly defined
chemically.

Other prices given in the Statistical Abstract are those of
wheat, barley and oats, which are obtained by averaging the
records of sales in the various corn mai'kets of the country.
The prices of minerals, including pig-iron, at the places of
their production, can be obtained approximately by dividing
their estimated value by the number of tons produced.

5. Index-numbers. — When measurable phenomena (such
as prices or wages) are influenced (1) hy causes special to
particular instances, (2) by general causes presumably acting
on all the phenomena, it is important to disentangle tfie
general causes from the special. Thus the ptice of wheat is
influenced by the weather, acreage under the crop, and the
harvests in all the wheat-growing countries ; the price of coal
by the fluctuations in demand : these are causes special to
these commodities. The prices of wheat, coal and all com-
modities are influenced by the relation of the amount of
money and its substitutes to the work that has to be done
by them : these are general causes. To determine the effect
of the general causes, that is, to determine the general change
of price, which varies inversely as the purchasing power of
gold, it is necessaiy to eliminate special causes. TIjis is done
by averaging together the price changes shuwn for a number
of different commodities, as follows —

* Even then exceptional years like 1900, when there was a great demand
for the coal of South Wales, should be excluded.
K

130 AJSr ELEMENTARY MANUAL OF STATISTICS

As many commodities are taken as possible, for which a
perfectly definite price quotation is current, great care being
taken to avoid changes of quality ; in practice, the number
of such commodities is not great, and retail prices must
generally be ignored. The average price of a period of years
is taken as base, and equated to 100; the prices of other
years are then expressed as percentages : e. g. from the table
above, we should have if we took 1870-79 as base —

Prices.

Proportionate numbers.

Average 1870-79 .

Year 1890 . . .

„ 1908 . . .

Wheat.

11-5

7-8

8-4

Sugar.
31-9
16-3
13-0

Tea.

16-3

10-6

8-0

Wheat.

100

Sugar.

100

Tea.

100
65
49

The average of the numbers so found for any year is the
index number for that year. This is very nearly the method
employed by Mr. Sauerbeck to obtain the index-numbers*
given in Column 12.

Another method is to follow the process used in the last
chapter (p. 119), thus — Imports in 1890 were valued at £356.
At the prices of 1902 they would have been worth £324.
Prices in 1890 were therefore higher than in 1902 in the
ratio 356 : 324 = 110 : 100. The import index-number for
1890 is therefore 110, when 100 is taken for 1902.

The first of these methods assigns equal importance to
each of the commodities chosen, at their average price in the
base-period.f To measure the abstract quantity "change of
purchasing power " or " appreciation of gold " one commodity
is as good as another, and one kind of average is as good as
another ; it is only necessary to take a sufficient number of
commodities to allow the laws of averages free play.

* To facilitate comparison his index for 1902 is equated to 100, and
the rest of the numbers raised in proportion.

t A change of base-period may afiect the arithmetical importance of
special commodities in the average, as may be seen by taking the three
commodities used above and taking 1908 as base ; but the difference dis-
appears when many commodities are taken, unless abnormal years are
deliberately chosen.

PRICES 131

The second method is more objective or concrete; it is
used to find the value of a definite group of commodities,
and this could be done exactly if the data were sufficient.
In such cases the method of index-numbers is only a method
of abbreviating computation and overcoming the absence of
complete information. It is justified, when it can be shown
by the principles of averages that the correct objective result
must be approximately reached. Similarly, if we wish to
find the change over a period in average wages of several
groups combined, we could, if we bad complete information,
work out the actual average year by year; but, in fact, we
can only find the ratio changes for the various groups and
have to combine these into a wage index-number by the use
of suitable weights. In fact, the method suitable for concrete
index-numbers differs from that convenient for abstract index-
numbers chiefly because weights must be used for the former
(unless it can be shown that they would not affect the result),
while they can be very often ignored with the latter.

Consideration will, in fact, show that either of these
methods is equivalent to comparing weighted averages of the
prices. It was stated and partly proved on pp. 18 and 32 above
that errors involved in such a process tended to neutralize
each other; supposing there to be one ideal true method,
all others may be regarded as differing from it by the intro-
duction of many minor errors. Experience shows abundantly
that many different methods of computing price index-
numbers yield approximately the same result, when proper
care is taken to avoid biassed and preponderant errors.

6. Index-numbers may, then, be taken on authority by
those who do not desire to follow the extremely interesting
analysis on which various methods of obtaining them are
based,* as showing with approximate correctness the general
change of prices of the group of commodities to which they
relate. Sauerbeck's numbers are typical of wholesale prices

* Among many books and articles on the subject, the reader may be
referred to Chapter XVIII of A Plain Guide to Investment and Finance,
by Mr. T. E. Young, in this series.

E 2

132 AN ELEMENTARY MANUAL OF STATISTICS

of raw materials in the United Kingdom, the export numbers
of prices of those commodities (principally manufactured
goods) which are exported, the import numbers of the great
variety of food, raw materials and other commodities imported.
If a diagram is made of these three series, their general
resemblance will be marked, and it is an interesting exercise
to trace the dates and examine the causes of the differences.
It is very important to define the group from which the
sample prices are selected. It is to be observed that none of
those in the table relate to retail prices, nor is rent or the
cost of labour involved. They do not represent the inverse*
of the value of gold in the hands of the consumer.

7. No study of statistical records involving price or value
is complete without reference to the general change in the
purchasing power of gold thus indicated. The changes may
thus be described since 1855 : The years 1872-3 were a time
of great price inflation, otherwise the general prices fluctuated
about the same general level from 1855 to 1874 ; from 1874
to 1896 prices fell enormously, and but for slight recoveries
in 1879-80 and 1887-9, almost continuously, the ratio of
prices in 1874 and 1896 being 3:2; from 1896 onwards a
considerable recovery took place, including two sharp inflations
and corresponding falls about 1900 and 1904-8.

8. In dealing with the statistics of Eastern and South
American countries, it must be remembered that in many
cases statistics are given in silver currency, whose value in
terms of gold fluctuates with the price of silver shown in the
table, p. 128. The Statistical Abstract for Foreign Countries
(published annually by the Board of Trade) contains the
information necessary for converting the values.

* If prices fall, the value of gold in terms of commodities rises, and
conversely. '

CHAPTER V

PRODUCTION

1. The more advanced the stage of manufacture, the
further removed from the raw material, the more difficult
it is to measure the qimntity produced by an industry, and the
scarcer are the statistics of value. We have generally to be
content with statistics of the quantity of raw material used,
and the value of that part of the completed goods which are
exported; for no general record is kept of goods produced for
the home market except in the instances given in the next
three paragraphs. The Board of Trade is now engaged in
working out the results of a recent Census of Production,
from which we shall for the first time have general statistics
on the subject.

Part I of the Preliminary Tables was issued in October,
1909 ; this contains statistics for Coal Mines, Iron and Steel
and Tinplate Works and Textile Factories. The following
table illustrates the information given —

Cotton Factories op the United Kinodom, 1907.

Output. Quantity. Value.

OOOOOO's

Yam 1530 lbs. £80

Piece-goods .... 7091 yards 82

Other — 15

Total £177

Cost of materials, etc £130

Excess of value of output . ... £47

Number of persons employed —

Wage earners 560,478

Salaried persons .... 12,391
133

134 AN ELEMENTARY MANUAL OF STATISTICS

The intention is to show for each industry the value
added to the materials by the processes of that industry.
We do not yet know in what detail these statistics will be
published.

2. The Board of Agriculture and Fisheries for Great
Britain collects statistics, which are based on local estimates
and not on direct enumeration, as to the amount of land
devoted to various uses, and the estimated yield year by
year of the various crops. These are published in the
Board's Annual Agricultural Statistics, the Department of
Agriculture and Technical Instruction of Ireland also publishes
an Annual Report, and the results are summarized in
considerable detail in the Statistical Abstract. There are
no general statistics of the production of wood, fruit, meat or
dairy produce, but there are estimates of the numbers of
horses, cattle, sheep and pigs. From these latter, combined
with expert investigation, estimates have been made * of the
quantities of meat and dairy produce produced, imported and
consumed. There is no estimate of the total value of the
agricultural produce of the country. The weight and value
of sea fish landed is estimated year by year.

3. The statistics as to minerals are good and complete,
owing to the fact that mines have long been subject to
inspection ; Part III of the General Report on Mines and
Quarries f deals with the quantity and the value of the
output of coal, copper, lead, tin, zinc and other metals.
These figures are summarized in the Statistical Abstract,
which also gives the production of " pig-iron " (the first form
in which the metal is obtained from the ore, and the raw
material of the iron industries and of a great part of the
production of steel) both from British and from foreign ores.
The amount of these minerals consumed at home can be
obtained by adding the imports to, and subtracting the
exports from, the home production, in the case of coal and

* Statistical Jcrwmal: Eew, 1892; Crawford, 1895, 1899; Craigie,
1903; Hooker, 1909.
t M.'_g. Cd. 4343 of 1908.

PRODUCTION

135

pig-iron ; for other metals special knowledge is needed, since
both metal and ore are imported.

The production of pig-iron and steel * is estimated as
follows —

Pig-iron.

Steel.

United Ejngdom.

United Kingdom.

OOO's tons.

1891-

-1895

average

7,040

3,030

1896-

-1900

8,890

4,590

1901-

-1905

8,769

5,125

1906

10,109

6,462

1907

10,114

6,522

4. The number and net tonnage of ships built in the
United Kingdom is given in the Statistical Abstract, and
distinction is made between ships sold to foreign countries
(for mercantile or naval purposes) and those retained under
home ownership. (The numbers built for the Royal Navy
are not given.) There are great and rapid fluctuations in
these totals, and those for single years should never be used
in isolation.

5. In the case of the textile trades we can obtain
comparative measurements from the raw material, and some
further ideas of their progress and importance from the
Factory Inspector's statistics of the number of persons
employed. The import statistics of cotton, j ute and silk show,
when re-exports are subtracted, the amounts of these fibres
brought into the home market each year ; and trade journals
show more accurately the amount actually used. For wool,
the sources are numerous, and the amounts used are shown
in the following extract (see next page) from the annual
report of the Bradford Chamber of Commerce.

Of this wool, some is exported after combing, some after
spinning, some as cloth or carpets, some as clothes; the
remainder is worked up and used at home. Also some

* See H. of 0. 376 of 1908.

136 AN ELEMENTARY MANUAL OP STATISTICS

Supply of Wool, etc. (Million lbs.)

Foreign and
Colonial
Wools im-
ported and
retained.

Home-grown

Wools (total

clip less

exports).

Wool from

imported

skins.

Shoddy,
etc.

Total
quantity
of Wool,
etc., used.

Estimated
value.

£000,000's.

1880-4 average

235

119

123

497

1885-9 „

301

112

101

539

1890-4 „

343

130

118

623

1895-9 „

405

115

132

685

1900-4 „

380

105

145

659

1905

371

180

682

1906

417

100

190

739

1907

489

100

210

833

1908

427

180

741

foreign yarn is imported. We cannot then say that the
total production is proportional to the wool used.*

In the case of wool the value of the output for the home
market is probably rather greater than for the foreign. In
the case of cotton the value of exports is supposed to be
about foui-fifths of the total value. Thus a very rough idea
can be obtained of the value of the total product, and the
value added in manufacture can be estimated by subtracting
that of the raw material.

6. For other industries we have only the incomplete
indices afforded by the amount of raw material imported, and
by the value and quantity of manufactures exported ; since
in most cases some of the raw material is produced at home
and a great quantity of the manufactures are used at home,
and since the proportions of home and foreign production
and consumption vary, it is generally prudent not to base
any conclusions as to production on such imperfect data. In
the case of the manufactures which contain iron or steel,
however, the very considerable increase in the weight and
value exported, as shown in the Statistical Abstract, should
be noticed.

* See Economic Joumul, 1905, pp. 584-590, for statistics as to the
growth of the woollen industry.

CHAPTER VI

WAGES

1. Statistics of wages are very plentiful ; but there are
so many different ways of reckoning and paying wages, and
such diversity in the methods of stating rates of wages, that
these statistics are extremely difficult to handle, and give rise
to many misunderstandings.

Wages may be paid by time or by piece. In the former
case the rates of wages are so much per hour, per week or other
period. The payment does not nominally depend on how
much work is done, but there is very often an understanding
as to what constitutes an hour's or a day's work, or as to how
long a particular job should take. Rates of time wages are
very generally agreed on between employers and Trade
Unions, and when this is the case there is usually no diffi-
culty in ascertaining them. There is generally also an agree-
ment as to the number of hours which constitute a week's
work ; the recognized payment for this number of hours is
known as the wages for a " normal week," and this payment
is the rate generally quoted. It should be observed that in
those industries which are carried on at a disadvantage by
artificial light the " normal week " is shorter in winter than
in summer. Where, as in the building trades, payment is
by the hour, there is no certainty that a man will obtain
employment for the whole week, and in any case there is loss
of time in changing from one job to another in outdoor build-
ing work. In this case wage statements generally give the
rate per hour and the number of hours which constitute a
full week's work season by season. Overtime, that is time
outside the scheduled hours that constitute the normal week,
is generally paid for at a higher rate. In some trades over-

137

138 AN ELEMENTAEY MANUAL OF STATISTICS

time is so frequent as to make an important difference in
average earnings ; in others undertime is common. Besides
the week's wage there are in many cases bonuses for regularity
or rapidity of work, special rates for special work {e.g. harvest-
ing), payments other than money, as when an agricultural
labourer has a house at a cheap rate or land to cultivate for
himself or perquisites of any kind, or a coal-miner obtains
house-coal at a low price. It is thus necessary to have special
knowledge of the conditions of employment in each trade
before using the bare statements of weekly rates of wages.

2. Piece-rates are, of course, rates of payment for the
performance of defined tasks. They very frequently are
arranged between employers and employed in the form of
elaborate piece-lists (or price-lists, as they are frequently
called), which define the exact nature of the task and show
innumerable variations of payment corresponding to the
various peculiarities of material or machinery by which the
work is lightened or made more arduous. The " prices " are
usually arranged with a view to the amount of work an
ordinary man can do at ordinary pressure in a normal week,
so that the week's earnings shall depend rather on the skill
or vigour required than on the accident of the special job.
Thus printers are paid at a higher rate the smaller the type
used. Coal-hewers are paid more per ton when working in
narrow seams than where the coal is more easily obtained.
Weavers are paid more per yard woven for every additional
complexity of the loom. In the large, thoroughly organized
industries, especially in the cotton manufacture and in
mining, this equalization of earnings is carried to an extra-
ordinary complexity, and the lists are frequently adjusted to
suit new conditions as they arise. It is obvious that a piece-
list in itself does not give any information as to earnings.

There is in reality no well-defined distinction between pay-
ment by piece and payment by time, for time-rates often
imply a definite amount of work, and piece-rates are often
arranged to produce a definite total of earnings. In fact,
there are many methods of payment which are partly on a

WAGES 139

time- partly on a piece-basis; for example, in engineering,
when time-rates are paid, a definite number of hours is
sometimes allotted to a job, these hours are paid for however
rapidly it is done, and in addition a bonus paid for rapid
work.* The distinction between the methods of payment is,
however, of statistical importance, for it is much easier to
obtain correct accounts of the week's earnings when on a
time-basis, for the statements are given in the form wanted,
and there is little variation from man to man ; while earnings
on piece-rates vary greatly according to skill, opportunity and
energy, and information has to be obtained by special
inquiries as to individual earnings firm by firm ; further, in
many occupations on a piece-basis the hours of work vary
from man to man and from week to week.

3. Changes of piece-rates are, in the larger industries, at
any rate, made by a general percentage increase or decrease
of the rates paid to many classes of operatives at once. Thus
on June 1, 1909, the rates for about 190,000 coal-miners in
South Wales and Monmouthshire were decreased "7^ per
cent., leaving wages 47| per cent, above the standard of 1879 " ;
that is to say, before the change rates were 55 % above those
which were arranged in 1879, and have been modified in
various details since ; the reduction was 7^ /^ off these 1879
rates, but 7^ on 147|, i.e. only 5 % (nearly), on the rates
immediately before the change. Where the current rates
differ greatly from the standard, it is very important to know
on what basis the change is reckoned.

It by no means follows in this case that the ordinary earn-
ings in June 1909 were exactly 47|/^ higher than those
in 1879. Rates may remain stationary while facilities for
production improve, or while the normal week is shortened.
In the cotton industry, in particular, slight improvements
or alterations of machinery are continually being made, which
result in greater productiveness by the operatives, with or
without additional intensity of work. Sometimes a nominal

* For illustrations [of these methods see Methods of Industrial
Memimeration, Schloss, 1892.

140 AN ELEMENTARY MANUAL OF STATISTICS

reduction is exactly counterbalanced, so far as the week's
earnings are concerned, by increased ease of production. In
all industries where piece-rates are common it is necessary to
make occasional inquiries as to actual earnings under ordinary
conditions in a normal week, to check the results shown by
the percentage changes of rates. These changes do not
always show the exact date nor even the approximate amount
of the resulting changes in earnings, for the operatives
often try to make the same earnings after a reduction as
before by more intense application, or neutralize an increase
by more leisurely work, and it is only after au interval that
the earnings settle to a new level. Making allowance for
these tendencies, it remains true that in general earnings for
a full week change nearly at the time of the published dates
of change of rates, and roughly in proportion to these when
they are considerable.

4. In the case of time-rates, the information available is
easier to use. The rates generally quoted are those recognized
both by the Trade Unions and by the Masters' Associations (if
both are effective bodies), and changes are the result of
public negotiations. The Trade Union rate is a minimum
below which no member of the union is allowed to accept
employment ; in trades where the union is strong this regula-
tion is actually followed throughout large districts, and even
non-union men are unlikely to receive less ; where the union is
weak the so-called minimum may be a rate which the work-
men wish to get recognized, while many are in fact working
for less. Special knowledge is, of course, necessary for each
trade and district before the actual significance of the rates
can be known. It is often supposed that the Trade Union
rate is a maximum as well as a minimum; this is not the
case in those important industries where there is scope for
skill and intelligence ; wage-sheets show that payments range
several shillings a week above the minimum rate. In fact, it
should never be assumed that the Trade Union minimum is
the average, or that it bears the same relation to the average
over a series of years. As in the case of piece-rates actual

WAGES 141

inquiries as to earnings must be made from time to time to
correct the impression given by detailed statements of changes.

5. Changes in earnings take place also in many other
ways. Where, as in the case of railways or the police, the
men are graded and promoted from grade to grade, or receive
additional payment in the same grade as their period of
service lengthens, a change can be made by an acceleration
of promotion or of increase, without any change in the
schedule of rates. Where processes of manufacture are
changing, it may easily happen that the rates fixed for work
at new kinds of machinery result in earnings above or below
those made formerly by the operatives who tend them. Such
changes are continually taking place in all mechanical in-
dustries, and the whole manufacture and the relative numbers
at various wage-levels may be revolutionized without a single
change of rates taking place. This is only one aspect of a
wider process ; for in a progressive country some industries
are always growing and new industries introduced, while
others are stationary or decaying; young persons enter the
former trades and find no opening in the latter, and so the
population shifts imperceptibly from industry to industry.
This tendency results on the whole in an increase in average
earnings of the working-class as a group, over and above that
shown by changes of earnings in particular industries. Such
changes, whether within an industry or in all industries
together, can only be measured by occasional complete in-
quiries as to earnings, combined with estimates of the
numbers employed.

6. The official information as to rates of wages is as
follows : The Labour Department issues from time to time
statements of the time-rates recognized in several industries
and in many districts, and also publishes abridgements of
price-lists and sliding-scales * in force. The time rates are

* Sliding-scales are arrangements (formerly prevalent and still existing
in the coal and iron and steel industries) by which recognized rates change
by defined amounts in accordance with the rise or fall of the prices
realized for the products of an industry. They have in recent years
been superseded in important cases by other methods of adjustment.

142 AN ELEMENTARY MANUAL OF STATISTICS

summarized in the annual report of the Department. A
report " on changes in rates of wages and hours of labour " is
published annually, recapitulating and summarizing the
details shown monthly in the Labour Gazette. More general
inquiries (or censuses of wages) were made as to earnings in
the years 1886 and 1906 ; the results of the former were
published in a series of volumes from 1889 to 1893, those of
the latter are, or will be, contained in reports of which the
second was issued in September 1909. There have also been
special reports on Agricultural Wages in 1900 and 1905. A
great part of what is known ofBcially as to general changes
of wages is printed and discussed in the three series oi Memo-
randa relating to British and Foreign Trade and Indvstrial
Conditions, generally known as the " Fiscal Blue Books " (Cd.
1761, 2337, 4954). The annual publications are full of im-
portant information, but they give no data as to the changes
indicated in paragraph 5 on the preceding page ; in fact, such
changes are ostentatiously ignored. The wage census of 1906
will, it may be hoped, when used in conjunction with occu-
pation statistics (see p. 91 above), make possible a general view
of the result of changes of all kinds during 20 years ; but it
will certainly not be an easy matter to make the comparison.
7. The following are examples of the information as to
changes of wages and hours tabulated by the Labour
Department : —

No. of

Locality.

Occupation.

Date of change
in 1908.

work-
people
affected.

Particulars of changes.

1. Preston

Carpenters

Julyl

360

Advance of irf. per hour
(U. to 9Jd).

and joiners

2. North

Blastftimace-

1st making-

600

Decrease under sliding

Stafford-

men

up day in

scale of 3i%, leaving

shire

October

wages 19J% above the
standard.

3. London

Printers' as-

let pay in

2,400

Advance of Is. 6d. per

sistants (ro-

January

week.

tary ma-

chines)

WAGES

143

These would lead to the following items in a summary
table : —

Industry.

No. of persons
affected.

Net amount of changes
per week.

1. Building trades .

2. Pig-iron manufacture .

3. Pnnting, etc.

360

600

2,400

2s.* X 360 = £36.
9|dt X 600 = £24.
la. 6d. X 2,400 = £180.

* Assuming that 48 hours is the length of the normal week averaged through the seasons.
The summer hours are given as 49^.
t Assuming, for the sake of the calculation, that the standard which is not stated is 25s.

The complete summary for 1908 is as follows : —
Changes of Rates of Waoes in 1908.

Groups of trades.

No. of workpeople
affected.

Net amount of change per

week in the wages of

those affected.

By increases.

By decreases.

Increases.

Decreases.

Building trades .

9,110

3,091

226

—

Coal-mining.

100

283,150

—

47,085

Other mining (iron,

etc.) ....

14,220

—

2,936

Quarrying .

2,661

—

290

Pig-iron and iron and

steel manufacture

69,247

—

9,656

Engineering and ship-

building trades

19,931

83,531

—

4,050

Other metal trades

219

1,654

. —

207

Textile trades

12,026

734

1,132

—

Clothing trades .

2,112

183

149

—

Printing, etc., trades .

10,808

956

—

Glass, brick, etc., trades

5,637

1,864

208

—

Other trades

2,703

3,678

—

368

Employees of Public

Authorities

56,543

191

2,750

—

Total .

119,327

464,216

Net Decrease
59,171

The numbers are exclusive of 379,790 persons in coal-
mining and the iron and steel manufacture, whose wages rose
and fell by equal amounts in the year. Individuals are only
counted once, however often their wages were changed.

144 AN ELEMENTAEY MANUAL OF STATISTICS

Chaiiges in rates of agricultural labourers, seamen, and
railway servants are not included in this table, but are the
subject of separate analyses further on in the report.

8. It is noticeable how largely coal-mining wages account
for the totals. This has been generally the case since the
beginning of these records. Wages in the coal industry and
in the manufacture of iron and steel change frequently, depend-
ing as they do in many cases on the ascertained selling prices
of the products. The wages fluctuate more widely than
wages in general, and the changes are in no way typical
of changes of average wages in the whole sphere of industry.
Unfortunately these changes are the most obvious, and are
frequently quoted as of much greater importance than they
are. The actual rise and fall in these special industries can
only be ascertained by observing them over the long period
of the ebb and flow of industry. The table on page 145 shows
the registered changes from the beginning of the series.

Thus the net increase in the week's wage bill shown by
these changes is only £274,000 in 16 years, while the
difference between the minimum of 1895 and the maximum
of 1900 was only £438,000, and apart from mining, this in-
crease was only £] 50,000. Now the total of wages other than
agriculture in the United Kingdom is probably £13,000,000
per week or more. It is incredible that changes should have
been so slight, and it seems certain that the totals of these
registered changes cannot be properly applied to the total
wages bill.*

The use of these records is of a less general nature ; when
mining is subtracted, the remainder shows in what years
wages were rising and when falling, and to some extent when
the movement was rapid and when slow. The more detailed
statements relating to separate occupations in separate towns
are of the greatest use in making it possible to keep the
records of time- and piece-rates up to date.

* In any case, the total wage bill would grow about 1% per annum
from the increase of the population, and this would be additive to any
increase shown in the tables above.

WAGES

145

Net Gain or Loss to

Weekly Wages Yeah

BY Year.

Mining and
quarrying.

Pig-iron and

iron and
steel manu-
facturers.

Textile
industries.

Other

industries.

Total.

1893

Gain.

£0

Loss.

»'3

Gain.
£0

Loss.
30's

Gain.

£0

Loss,
)0's

Gain.

Loss,

ao's

Gain.
£01
13

Loss.
lO's

1894

—

1895

—

1896

—

1897

—

1898

—

1899

—

1900

168

209

1901

—

1902

—

1903

—

. —

—

1904

—

1905

—

1906

—

1907

176

—

201

—

1908

—

1909*

—

Result in 16 years
11 months

Ill

—

274

—

* First eleven months.

9. The course of wages, whether in separate industries or
as a whole, has been studied by the help of these detailed
changes together with similar information in earlier times as
to Trade Union rates and the changes in piece-rates. These
data have been pieced together with the help of evidence
and estimates as to actual earnings drawn from a great
variety of sources in a series of articles in the Journal of the
Royal Statistical Society by Mr. G. H. Wood and the present
author, of which the first appeared in 1895. All results are
tentative till we have the wage census of 1906, but there is
sufficient evidence to support the statements of the following
table, as showing the general movements of rates of wages
with fair accuracy. It is to be remarked that in the long
run wages for work of any particular grade of skill approxi-
L

146 AN ELEMENTARY MANUAL OF STATISTICS

mate to each other, so that a sample which includes the
most populous industries must be fairly typical of industries
all together.

The working up the data is actually accomplished by
means of index-numbers on a basis generally similar to that
of price index-numbers, but the details are more complicated
and too technical for discussion here. The principle is to
take as data the changes recorded, which can be ascertained,
rather than the actual earnings, which can be stated in many
dififerent ways according to the bias of the informant. Thus
in the table on page 147 the average wage in each industry
is taken as 100 in 1880, and the estimated average for
other years is given as a percentage of the average in this
standard year.*

The column headed "general" shows the course of the
average of the wages of all adults employed in all the indus-
tries for which the necessary calculations have been made
(including the four groups in the following columns), allowing
for the shifting from one industry to another and from grade
to grade within the industries. The following four columns
show similar figures for four important industrial groups.
The last column shows the unweighted average (that is, the
average of certain rates without reference either to the
numerical importance of the different industries, or to the
relative growth of some industries) as given in the " Fiscal
Blue Books" (Cd. 1761 and 2337), and in their continuation
in Od. 4954, published in Novemlaer 1909.

The general conclusion from any of these columns is that
wages were nearly stationary from 1880 to 1887, rose rapidly
from 1887 to 1891, were again stationary till 1897, rose
rapidly to 1900, fell very slowly till 1905 back to the level of
1899, rose again in 1906-7, and show a downward tendency
in 1908. Wages at the maximum of 1907 were higher than
in 1900, and considerably higher than at any previous
date.

* 1880 is taken aimplyforconvenienee of working. The results shown
do not depend at all on what year is taken as standard.

WAGES

Index-numbees of Average Eates of "Wages.

147

Years.

100

■<

•Eb

Board of Trade

Unweighted

Average.

1880

100

1881

100

104

100

103

102

1882

103

104

100

105

103

1883

103

105

100

105

103

1884

103

105

100

104

102

1885

101

104

100

103

100

1886

100

103

100

1887

101

104

101

1888

104

108

101

104

102

1889

110

108

103

108

105

1890

114

111

100

104

111

108

1891

115

113

100

104

111

110

1892

115

100

105

109

108

1893

115

107

108

1894

115

107

108

1895

115

116

108

107

1896

115

lie

109

111

108

1897

116

111

113

109

1898

120

116

101

112

116

112

1899

123

120

103

113

119

1900

130

123

109

115

119

120

1901

128

123

110

115

119

118

1902

126

123

110

115

118

116

1903

125

123

110

115

117

115

1904

123

110

115

117

115

1905

123

127

110

115

117

115

1906

126

127

110

115

119

117

1907

133

131

110

115

119

122

1908

Falling
slightly

131

110

115

117

121

On the same basis the previous maxima and minima for
the general average were about —

Index-number.

1850 68

1855 79

1858 75

1866 90

1868 ... .... 87

1874 106

1879 . . 99

L 2

148 AN ELEMENTARY MANUAL OP STATISTICS

The numbers in this last table are computed from Mr.
G. H. Wood's table, pp. 102-3 of the Statistical Joiornal, 1909.

10. All the statistics of the preceding paragraphs (8 and 9)
refer to rates of money wages of persons working full time in
a normal week, excluding casual workmen and others not
regularly attached to a definite trade. They refer mainly to
men, but include the very large numbers of women employed
in the textile industries. Two important adjustments must
be made before they are applied to measure the economic
well-being of the working-class, one for unemployment, the
other for the change in the purchasing power of money. The
following chapter shows that employment is more regular
when wages are rising and vice versd, and that over a long
period unemployment in such a group of industries as those
considered has neither increased nor diminished perceptibly.
The effect of allowing for unemployment would therefore be
to increase the fluctuations without affecting the trend of the
series shown in the first column of the table on p. 147.

As regards purchasing power in retail commodities, it was
stated in Chapter IV above, that the measurement was very
difficult ; in fact, authorities do not agree as to the movement
of prices, especially when rent is included. The following
table shows the results of a calculation by the present author.*

Rates of
money wages.

Prices.

"Real" wages. t

1852-1870

Rising fast

Rising

Rising consider-
ably in the
whole period

1870-1873

Rising very fast

Rising fast

1873-1879

Falling fast

Nearly stationary

1879-1887

Nearlystationary

Falling

Rising

1887-1892

Rising

Rising & falling

Rising

1892-1897

Nearlystationary

Falling

Rising

1897-1900

Rising fast

Rising

1900-1904

Falling a little

Falling & rising

Stationary

* From Appendix to the Dictionary of Political Economy, p. 801.
t For another view see Mr. Wood's article just quoted (Statistical
Joitmal, March 1909).

WAGES

149

CD
O

In
O

l-t

pi?
n

Q
H

•5

-B -* M CO O

1-1 -* l-H oq

M O 1> 00

rH l-H

, • 05 00 i-< O

lO M 00 00

00 CO 05 vO

« CT ^ ^ rH

oq r-l

oq i-H

60«.
and
over.

"^ 1 1 1

T' 1 I 1

•?" 1 1 1

■^111

oq M 1

•« 1 1 1

■gt-

p s ■
"•OS

■? 1 1 1

'^ \ \ \

*? 1 1 1

.-gS

>o 1 1 1

1 1 1

oq 1 1 1

lA P

"2 i.

9 S .

"?> 1 1 1

■^111

9 1 1 1

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00 1 1 1

oq M 1

>o 1 11

o 3

■*

"3 K

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"^ 1 i 1

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00 ! 11

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'¥"r' 1 1

CO CO 1 1

«• eg

oq 1^ ' 1

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OJ ill i 1

o P

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"Sk

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M O r-l vp

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i-H M oq <N

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6 CD ■

6 ii oq ■

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t~ n Oi CO

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■* -^ iln ill

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i-H m oq r-H

r-i oq

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r'T-O'?

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oq ill o M

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oq -^ TO

lO oq G<i

"* oq

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1 p t~ 05

1 >p o ^

1 Oi 00 1^

5§"

1 lO ill r-

1 >h « ■*

1 CO CD ill

oq M

rH -^lO

oq Tji lo

■O ag

. ^ cp n

1 00 CO 00

I o oq Tf

1 ■ ill -*

\ ' Oi t~

1 oq o 05

l-H i-H

1—1 1— 1

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....

1 ■ • ■ •

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S TS

« rjj

fl rt

5 rt «

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<! . a) eS .

• S 0!

150 AN ELEMENTARY MANUAL OF STATISTICS

The prices included are principally those of food. " Real "
wages mean wages expressed in terms of commodities, that is,
money wages corrected for change in purchasing power. It
is very noticeable that periods of rapid increase of wages have
been those also of rising prices, which have neutralized to
some extent the benefit of the wage-increase ; and that periods
of stationary wages have been those of falling prices, which
have had practically the same effect as an increase of wages
with an unchanged price.

11. All the preceding figures apply to averages, not to
individual persons. We have extremely few records of the
earnings of individuals for periods longer than a week, though
information of a difficult and complex nature is accumulating
as to the number of weeks' work and the amount of overtime
or lost time obtained or obtainable in a year in various
occupations, and the variation from year to year. On the
other hand, we are now learning from the Wage Census of
1906, the relation of the weekly wages and earnings of
individuals to the average. The table on p. 149 shows in
abstract form the kind of information obtained. The earnings
are those of all persons, whether working full time, overtime,
or short time. The men earning less than 15s. were in most
cases on short time, and were possibly in a few instances
earning money also in other places. The boys and girls
earning less than 5s. in the cotton and woollen industries
were generally half-timers. The statistics refer to the returns
obtained from all the principal districts for these industries
in the United Kingdom. Tables showing the earnings of
those who worked the normal week are also given in the
Reports (Cd. 4545 and 4844).

CHAPTER VII

EMPLOYMENT

1. We are entirely dependent on the Labour Department
of the Board of Trade for statistics of the amount of employ-
ment and unemployment. As in the case of wages there is
practically no official information, except as to workpeople
regularly attached to a trade ; the conditions and numbers of
casual workers, of those in the so-called sweated trades, of
persons physically or intellectually incompetent to do regular
work, of habitual vagrants, and of others chronically un-
employed, under-employed, or unemployable, are not the
subject of official statistics; in fact, there is practically no
evidence, official or statistical or of other kinds, to show the
actual numbers in these classes, or to show whether these
numbers are stationary, decreasing or increasing.

In several occupations and industries employment (or, at
any rate, payment) is practically regular, modified only by
occasional overtime and by the temporary engagement of
extra hands in busy months. Among these are the Army,
the Navy, the government services, such as the Post Office,
municipal, and other local government services (police,
sanitation, etc.), and the railways, and perhaps other land
transport services. Nearly as regular as these are domestic
service, and the manufacture and distribution of ordinary
food and drink. There are thus large sections of the working
population who are not directly affected by the question of
unemployment.

Next to these come a group of industries, in which certainly
more than two million persons are employed, in which it is
the custom to regulate the working week in relation to the
demand for the product, employing nearly the same number

151

152 AN ELEMENTAEY MANUAL OF STATISTICS

of persons in good trade and in bad, but working short time
when the market becomes overstocked. In these cases the
employees suffer from reduced earnings, and may benefit by
some enforced leisure, but are not actually unemployed.
Coal-mining is the most conspicuous industry of this group.
The textile trades (cotton, wool, and others) organize employ-
ment with a similar result ; short time is worked, or the work
is spread out among the operatives, when the demand is
slack ; but the great number of those employed in moderately
busy times draw some wages nearly every week. In times
of pressure, however, the textile industries generally find it
possible to engage more operatives, whence it appears that
there must be a reserve array who are not regularly at work ;
part of this army consists of married women who have with-
drawn from lactory work, but return from time to time. Of
C(iur>e these industries gradually grow or diminish quite apart
from the fluctations betwet-n periods ot good and bad trade.

The ordinary statistics of unemployment do not relate to
either of these large groups. The Labour Gazette * gives every
month tables which indicate the amount of employment in
industries of the second group. Thus for mining and quarry-
ing the average number of days worked per week is given;
for the cotton, woollen and worsted, linen, jute, lace, hosiery,
silk, and boot and shoe industries, and some others, the
changes month by month in the number of persons employed,
and in the aggregate of wages paid, by a very large number
of firms, are shown in a series of tables. These tables are
not easy to use for the purpose of tracing changes in wages or
numbers over a long period, for returns are not obtained from
exactly the same firms throughout, and they do not include
the whole trade; but they show very clearly when trade is
declining or improving, they afford a check on other estimates
of wages, and, combined with statistics of output, indicate
how various industries meet the ebb and flow of demand.f

* Monthly, price Id.

t For- an interesting discussion of this point, and a general analysis
of unemployment statistics, see Unermloyment in Lancashire, Chapman
and Hallsworth, 1909.

EMPLOYMENT

153

2. For the manufacture of pig-iron and tinplates, for iron
and steel works, and for dock-labour, the Labour Gazette
shows the number of furnaces or mills in operation, or the
number of persons employed. In these industries bad times
are not met by shortening the week or spreading out the
work among the operatives, and persons are therefore actually
unemployed; but there are no distinct statistics of the
numbers so unemployed, and estimates would have to be
made, if it were possible, by noticing the maximum ever
employed, and taking the deficit from this maximum as the
number out of work. The statistics are hardly sufficiently
complete for this purpose.

There are no definite statistics for agriculture labour or
for the ready-made or " bespoke '' clothing trades, but verbal
accounts are given every month of the state of employment.
For seamen we have records of the number shipped at the
various ports.

3. The only important industrial groups not named in the

previous paragraphs are the printing, building, furnishing,

engineering and shipbuilding trades. The ordinary statistics

of unemployment include considerable numbers of these

groups.

" Labour Gazette," October 1909.*

Industries.

Membership of the

Unions from whom

returns were

obtained,

September 1909.

Number
unemployed at

end of
September 1909.

Percentage

of
Memliership
unemployed.

Building ....
Coal-mining.
Engineering
Shipbuilding
Other metal trades
Textiles ....
Paper, printing, and book-
binding ....
Woodworking and furniture
Miscellaneous

58,917

139,746

171,370

57,280

41,540

115,821

59,127
35,165
16,790

6,432
1,669
18,592
12,855
2,286
2,721

3,820

2,719

655

10-9

1-2

10-8

22-4

5-5

2-4

6-5
7-7
3-9

Total

695,720

51,749

7-4

* Supplemented by additional details furnished by the Department.

154 AN ELEMENTARY MANUAL OP STATISTICS

The numbers included from coal-mining and textiles
are an insignificant proportion of the aggregate in these
industries, and contribute little to the total of unemployment
thus measured.

Among the Trade Unions of the United Kingdom only
the minority, who pay allowances to their members when
out of work (" unemployed benefit "), keep a record of the
members unemployed. Reports are obtained from this
minority by the Labour Department of those who are on
the unemployed books of the various branches at the end
of each month, together with the membership of these
branches. The table just given is compiled directly from
these reports. The numbers do not iuclude persons on
strike, sick or superannuated, who draw other "benefits"
from the unions.

The numbers for the building trades depend only on
carpenters and plumbers. The Operative Bricklayers
Society has no unemployed benefit. In the winter months
carpenters, painters and plumbers have more employment
than those in other building operations, and the percentage
of unemployment for all the building occupations would be
higher in the winter than that shown in the returns. There
is also much under-employment, or lost time, in the building
trades, where the hourly system of engagement is prevalent,
which is not shown in this table.

On the other hand, the engineering and shipbuilding and
perhaps the printing trades are adequately represented.

The figures refer almost exclusively to artisans; labourers'
unions do not generally have unemployed benefit.

These returns are, therefore, merely a sample of the facts
of unemployment, and there is little reason for taking
the resulting percentage as applicable to industry as a
whole. It is sometimes supposed that labourers are more
frequently unemployed than artisans; but this is not the
case when they are attached to industries in which skilled
work is prevalent, for the whole group, men and women,
boys and girls, skilled and unskilled co-operate, and the

EMPLOYMENT 155

labourers cannot stop unless the work is stopped. Agri-
cultural labourers obtain regular work if attached to a
farm, and those who do seasonal work find much the same
demand year after year. On the other hand, dock-labour
varies considerably.

Again, it has been shown above that many important
industries produce little unemployment.

The percentages shown by these returns can, then, only
be used to measv/re unemployment after a troublesome
and hazardous estimate; their use is rather to form an
iTidm of unemployment, which shall reach its maxima
and minima at the worst and best times respectively, and
fluctuate much or little as the state of the labour market
changes is unstable or steady. In the following paragraphs
the percentages dre used in this sense.

A study of the table on p. 156 shows that unemployment
has fluctuated in periods which are nearly decennial, the
worst years being 1858, 1868, 1879, 1886, 1893, and 1904,
and (probably) 1909; in the last two decades the periods
are less regular, a long spell of good employment (1886 to
1901) being followed by an abortive crisis in 1904, two fairly
good years in 1906, 1907, and bad years in 1908-9.

On the whole, it cannot be said that unemployment
as shown by these numbers has either increased or decreased
over a long period ; this would be seen better from a diagram
than from the averages given, for these depend very much
on what period is averaged.

The apparent severity of the worse periods arises from
the preponderance of the engineering and shipbuilding
trades, some branches of which fluctuate excessively. If
these industries are subtracted, the remainder only once
(1904) show a percentage higher than 6"1. The table on
p. 153 above, for October 1909, shows also in that month
for engineering and shipbuilding 13*6 °^ unemployed, and
for other industries 4'0 %. Column D on p. 156 shows the
effect of assuming that other industries as a whole are of
the same numerical importance as these two.

156 AN ELEMENTARY MANUAL OF STATISTICS

Index of Unemployment. Labour Department.
OF Trade Unionists Unemployed.

Percentage

All industries for

which Betnrns are

aTailsble.

Shipbuilding

and
Engineering.

Other
indus-
tries.

Decennial
averages.

Notes.

Ai.

Bi.

Aa.

Bs.

1851 . .

1852 . .

1853 . .

1854 . .
1866 . .
1866 . .
1867

1858 . .

1859 . .

3-9
6-0
1-7
2-9
5-4
4-7
6-0
11-9
3 '8

3-9
6-0
1-7
2-9
6-4
4-9
6-1
12-2
3-9

1-6
2-3
2-5
1-4

40
(9 years)

The numbers in the
Columns A are partly
based on the expendi.
tares on unemployed
benefits (Cd. 2337, p.
91).

1860 , .

1861 . .

1862 . .

1863 . .

1864 . .

1865 . .

1866 . .

1867 . .

1868 . .

1869 . .

1-9
6-2
8-4
6-0
2-7
2-1
3-3
7-4
7-9
6-7

—

1-9
5-5
9-0
6-7
3-0
2-4
3-9
91
10-0
8-9

1-8
1-9
3 1
2-7
■9
1-2
1-4
3-5
3-6
3-0

—

- 5-7

The numbers in Columns
B are obtained as ex-
plained in the previous
paragraph.

1870 . .

1871 . .

1872 . .

1873 . .

1874 . .

1875 . .

1876 . .

1877 . .

1878 . .

1879 . .

3-9
1-6
■9
1-2
1-7
2-4
3-7
4-7
6-8
11-4

—

4-4
1-3
•9
1-4
2-3
35
5-2
6-3
9
15-8

3-1

2-0

1-0

•9

1-6

2-5

3-5

■ 3-8

The numbers in Column
C are the result of fur-
ther information ft-om
certain Trade Unions,
and are given in the
labour Qazelte, January
1909.

1880 . .

1881 . .

1882 . .

1883 . .

1884 . .

1886 . .
1886

1887 . .

1888 . .
1889

5-5
3-6
2 '3
2-6
8-1
9-3
10-2
7-6
4-6
2 1

4-9
21

4-1
2

6-7
3-8
2-3
2-7
10-8
12-9
13-6
10-4
6-6
2-0

6-0
2-3

3-8
8-3
2-4
2-6
8-6
4-2
5-6
3-9
3-4
2-1

2-3

1-8

• 5-6

The numbers in Column
D are the simple aver-
ages of those in Columns
Bi and Ba, and are used
to reduce the over-pre-
ponderance of engineer-
ing and shipbuilding
in the unadjusted per-
centages (Cd. 2337, p.
92).

1890 . .

1891 . .

1892 . .

1893 . .

1894 . .
1896 . .

1896 . .

1897 . .

1898 . .

1899 . .

2-1
3-2
5-8

2 1
3-6
63
7-6
6-9
6-8
3-4
3-5
3-0
2-4

2'8
2-0

2-1
3-4
62
7-7
7-7
6-0
S'5
3-6
S'l
2-4

2-4
4-4
8-2

2-2
41
7-7
11-4
11-2
8-2
4-2
4-8
4
2-4

1-6
1-8

2-7

2
2-7
4-7
4-0
4-2
3-9
2-8
2-6
2-3
2-4

- 4-4

The averages In Column
E are from Columns A
andB.

1900 . .

1901 . .

1902 . .

1903 . .

1904 . .

1905 . .

1906 . .

1907 . .

1908 . .

1909 01
montha)

2-9
3-8
4-4
6-1
6-5
6-4
4-1
4-2
8-1

2-6
3-3
4-0
4-7
6-0
6-0
3-6
3-7
7-8

7-8

2-8
3-8
4-6
5-3
6-8
6-6
4-1
4-8
9 1

2-6
3-8
6-6
6-6
8-4
6-6
4-1
50
13-0

3-1
3-8
3-7
4-0
7-6
4-6
4-1
3-6
5-3

■ 6'2

EMPLOYMENT

157

It is important to notice that the more complete returns
now obtained reduce the percentage unemployed by about
•4. In the comparison of statistics subsequent to 1908 with
those of an earlier period, great care will be necessary.
The effect of the newer figures is shown in Column 0,
from 1898 onwards. The alteration from this adjustment
emphasizes the cautions already given as to the difficulty
in the use of these percentages in measuring unemploy-
ment.

Mean of the
years
1887-1895.

Mean number of members in dis-
trict, ex chiding superannuated .
Number of separate members un-
employed for as much as 3 days
some time during the year .
Percentage of membership
Average number unemployed at the
same time during the year .
Percentage of membership
Aggregate number of working days
lost through want of employment
Number per member
Average number of working days
lost by those members who were
unemployed for as much as 3
days in the year
Percentage number of members un-
employed during the year for —
Less than 3 days

3 days, and less than 4 weeks .

4 weeks, and less than 8 weeks
8 weeks, and less than 12 weeks
Over 12 weeks

1890.

1893.

6,344

6,934

1,356
21- J^

1,832

134
2-1

706
10-2

40,825
6-4

215,874
311

30-1

117-8

78-6

14-1

3-5

1-6

2-2

73-6
5-8
2-4
21

16-2

6,507

1,929
S9-7

403
6-1

123,166
18-7

631

70-4

129

4-6

2-8

9-3

5. The annual averages hitherto given are obtained for
each year by adding together the percentages shown at the
end of each month and dividing by 12, a method which
must give very nearly the same result as any more
refined calculation possible. If the general average is 4 % it
means that, so far as the group of occupations which are

158 AN ELEMENTARY MANUAL OF STATISTICS

included is concerned, one person in 25 is on the average
unemployed through the year.* This average may be
made up by one man in each 25 having no work in the
year, or each man losing one week in 25, or any distribution
of unemployment between these extremes. We have very
little information on this point, but the table on p. 157 (Cd.
2337, p. 101) shows the circumstances for the Amalgamated
Society of Engineers.

In this case it is seen that about one quarter of the
members bore the whole brunt of unemployment in the bad
year 1898.

6. There are good and bad seasons in the year in nearly
every occupation. These can be studied, if we average away
the peculiarities of particular years as is done in the following
table : —

Percentages of Unemployment, showing the Seasonal
Fluctuations.

(Compiled principally from the Xlth Abstract of Labour Statistics.)

Yearly
mean.

1-3

■<

!?;

•^

/1897

3-0

2-5

2-3

2-7

3-6

4-4

4-7

4-8

5-3

3-5

ims

6-0

4-4

3-1

2-9

2-7

2-6

2-8

2-6

2-3

2 '9

3-0

1899

3-0

2-6

2-5

2-2

2-6

2-3

2-8

2-4

2-3

2'2

2-6

2-4

All Trade

1900

2-7

2-9

2-3

2-6

2-4

2-6

2-7

3-6

8-8

3-2

4-0

2-9

Unions

1901

4-0

3-9

3-6

8-8

8-6

3-5

3-4

3-9

3'7

3-7

3'8

4-6

3-8

making

1902

4 '4

4-3

3-7

3-9

4-0

4-2

4-0

4-6

6-0

4-8

6-6

4-4

returns.

1908

5-1

4-8

4-3

4-1

4-0

4'9

6-5

5-8

6-8

6-0

6-7

5-1

1904

6-6

6-1

6-0

6-3

5-9

6-1

6-4

6-8

7-0

7-6

6-6

1906
Vl906

6-8

6-2

6-6

5-1

6-2

6-4

6-3

6-0

4-7

4-9

6-4

4-7

4-4

3-9

3-7

3-6

3-7

3-8

8-8

4-4

4-6

4-9

4-1

AverageB for 10 yeaiB
1897-1906.

Ml ... .

4-6

4-3

3-75

8-7

3'6S

3-8

3-76

4-1

4-3

4'35

4-3

4'9

4-1

Building (carpenters

and plumbers)

6-4

5'2

4-4

3 '6

3-4

3-7

3-5

3'3

3-8

4-4

4-8

6'9

4-3

Engineering

4-6

4-4

4-0

3-8

8'6

3-7

S-7

4-0

4-2

4-4

4-6

S'3

4-2

Shipbuilding .

8-0

6-8

6-1

6-3

6-2

6-1

6-8

8-1

9-2

9'9

7 '4

Printing .

4-8

4-3

3 6

4-2

4-5

4-4

3-7

6-4

4-6

2-9

4-2

4-4

Fumianing

7-9

6-6

3-2

8-2

4-1

4-2

4-6

4-9

7-1

4-6

* Not exactly, for persons losing only a day or two are not registered,
and, on the other hand, the names of those who get employment may not
be taken off at once.

EMPLOYMENT

159

Here we see a distinct but small monthly fluctuation when
all trades are merged together ; the months from March to
July are the best, December and January the worst. Decem-
ber or January is the worst month for the separate groups
except printing, while the best month varies from March to
August.

7. It is interesting and important to study these statistics in
times of depression in the light of these seasonal fluctuations.

1907.

a
1-3

i-s

Engineuriiig

8-2

2-S

2-9

2-8

2-9

3-4

.1-S

H-H

4-1

4-S

4-7

6-4

Shipbuilding

S-8

7-S

7-1

6-6

6-7

6-4

7-2

9-3

11-7

11-6

12-8

14-2

Others

3-9

3-r

S-3

3-0

3-1

3-4

3 3

3-4

S'8

3-8

4-2

4-8

All ... .

4-2

3-9

3-6

3 '3

3-4

3-6

8-7

4-0

4-6

4-7

5-0

6-1

1908.

Engineering

6-8

5-9

7-1

8-6

9-5

10-6

U-1

12-0

12-3

12-7

13-0

14-0

Shipbuilding

15-1

20-0

21-5

23-2

26-1

20-6

22-2

26-2

26-6

26-3

2S-2

24-7

Others

6-1

4-S

4-1)

5-2

5-4

6-K

5-3

5-7

,'j-4

s-.i

5-4

6-6

Ail ... .

6-2

6 '4

6-9

7-8

7-9

8 '2

8-2

8-9

9-4

9-5

9-1

9-1*

1909.

Engineering

13-1

12-8

12B

12-4

12-,5

12-1

11 '3

10-8

10-3

9-S

Shipbuilding

23-0

22-6

22-2

23-3

23-4

23-6

23-9

23-3

22-4

21-6

19-3

others

6-4

6-U

4-»

4-7

4-3

4-5

4-4

4-S

4-4

4-0

All ... .

8-7

«-4

«■«

«-2

7-9

7-J

7-9

7-7

7-4

7-1

6-6

* New basis of computation begins here.

These numbers were disturbed by strikes in the shipbuild-
ing and engineering trades, which were settled respectively
in May and September 1908; though persons actually on
strike are not included, the influence is always felt beyond
the nominal area. From and after December 1908, the
figures for "all" are on the new basis named on pp. 156-7
above, and are about '4 below the height they would other-
wise have reached ; a slighter eflect is produced in the other
lines.

It is very noticeable that the two industries of engineer-
ing and shipbuilding account for the apparent acuteness of
unemployment in 1908 and 1909. The maximum for

160 AN ELEMENTARY MANUAL OF STATISTICS

" others " is well under the percentage 7 '6 reached in
1904 (see p. 156).

In the previous table it is seen that the general per-
centage for all trades for November is equal to that for
October and less than that for December; but in 1908
there was a fall in November and no change in December,
hence October 1908 was relatively as well as absolutely the
worst month for employment in general. During 1909 there
was less than the usual fall in the early part of the year, but
August and onwards showed an improvement instead of the
usual seasonal increase of unemployment.

In engineering the changes from June 1908 to July 1909
are very nearly proportional to the ordinary seasonal change ;
no non-seasonal improvement is visible till August 1909. In
shipbuilding similarly the first definite sign of improvement
is in August 1909.

CHAPTER VIII

OTHER STATISTICS RELATING TO THE WORKING
CLASSES

1. Besides the statistics of occupation, production, wages
and employment already dealt with, there are several other
statements relating to the working-class, most of which are
summarized in the Anmoal Abstract of Labour Statistics.
We will omit the statistics of profit-sharing, of industrial
accidents, and of diseases of occupations, and deal briefly
with the tables relating to trade disputes. Trade Unions
and co-operation.

The statistics relating to strikes and lock-outs* are
obtained directly from the employers and Trades Unions
concerned during and at the end of the dispute. Apart
from information as to the wages and normal hours of
labour recognized before and after, and as to changes of
any kinds made in the conditions of employment or working
arrangements, the statistics collected relate to the causes
and to the results of the disputes and to the methods by
which they were terminated, to the number of persons
directly or indirectly affected, and to the number of working-
days lost.

By the number of persons directly affected is meant those
who are actually on strike or locked-out; in the number
iTidirectly affected are included " other workpeople employed
at the establishments where the dispute occurred, and thrown
out of work by the dispute." Clearly this latter category
is arbitrary ; if carpenters were the permanent servants of

* The information is given in much more detail in the Annual
Reports on Strikes and Lock-outs.
M 161

162 AN ELEMENTARY MANUAL OF STATISTICS

a firm whose works were closed they would bo classed as
" indirectly " affected, whereas if they were hired through a
contractor as required they would be equally affected by
the loss of work, but would not be included. In fact the
effect of a strike cannot be measured ; members of all
the industries, at home or abroad, who furnish material for
the manufactures which are stopped or use their finished
products, and at a later stage the great multitude of people
who in general provide the strikers with commodities which
they can no longer afford when their wages stop, are to a
greater or less extent thrown out of employment ; the effects
of a strike spread through industry like ripples over a pool
when a stone is dropped into it.

The number directly affected is rendered indefinite by the
difficulty in distinguishing them on any definition from those
indirectly effected. If weavers are on strike, the sizers and
dressers may cease work either because they sympathize with
the weavers' grievances, or because their work is useless
virhen the looms are stopped, or because the employer locks
out all hands. The effect is much the same, but in the
first case they are " directly," in the others " indirectly "
affected.

This difficulty of definition cannot be got over, and there-
fore the statistics, and others based on them, can only be
used as indication of the effect of disputes, and with due
caution for comparing one year with another.

The number of days lost through a dispute is computed
from the number employed at the beginning and the
duration of the stoppage of work. This is a little fictitious,
for there is no certainty that this number would have
obtained work throughout if there had been no stoppage,
and it is probable that either there will be extra work to
do after the dispute, or that more work has been done in
other places during the dispute, or that trade has been
permanently displaced. These criticisms have yet more
force when the loss of Wages is computed, as is sometimes
done unofficially in this country and officially in others.

STATISTICS OF THE WORKING CLASSES 163

In fact, the circumstances of strikes cannot be made the
subject of exact statistics; we can only note in general
terms whether they are becoming more or less acute as
the years go on. The following table shows the principal
statistics for the United Kingdom for 1893-1906.

Aggregate

No.
of

dis-
putes.

No. of
work-
people
directly
affected.

No. of
work-
people
indirectly
affected.

duration in
working-
days, i. e.
total number
of days lost
by persons

Percentage number of disputes
settled.

directly or

In favour

In favolff

Compro-

indirectly

of'work-

mised or

affected.*

people.

employers.

indefinite.

OOO's

0,000'3

1893

615

594

30,47

1894

929

257

9,53

1895

745

207

5,72

1896

926

148

3,75

1897

864

167

10,35

1898

711

201

15,29

1899

719

138

2,52

1900

648

135

3,15

1901

642

111

4,14

1902

442

117

140

3,48

1903

387

2,34

1904

355

1,48

1905

358

2,47

1906

486

158

3,03

1907

601
399

101

224

47
72

2,16
10,83

Not given

in this foru

1908

J for 1908.

' This includes in each year days lost by persons through disputes which began in a
previous year.

The high numbers of working-days lost were mainly due
in 1893 to the strike of coal-miners in the Federated Dis-
tricts, in 1894 to the Scottish coal-miners' dispute, in 1897-8
to the engineers' dispute, and in 1908 to the shipbuilders'
and South Wales coal-miners' and cotton operatives' dis-
putes.

The series in this table do not show any very definite trend,
nor any clear connection with the periods of good or bad trade
or of rising or falling wages.

2. The statistics relating to Trade Unions have been for

M 2

164 AN ELEMENTARY MANUAL OF STATISTICS

many years good and complete. In general, very careful
accounts are kept in detail of membership, receipts and
expenditure by the officials of the various unions, and are
published periodically for the information of their members.
These and special reports are easily available for the
Registrar-General of Friendly Societies and for the Labour
Department. The more interesting details are summarized
for 100 principal Trade Unions (with an aggregate member-
ship of 1,274,000 in 1906), and yet more information is
given separately for each of 25 societies (with an aggregate
membership of 972,000).

Number and

100 Principal Unions

membership
ofallUnions

from which

Expenditure on various

"benefits."

information

»H

is received.

1
§

1
.a

ODO'S

OOO's

£000's

£000'3

£000's

£000'

£000's

£000'8

1897

1298

1,659

1,089

2,231

1,971

331

647

256

147

114

318

1,891

1898

1267

1,684

1,068

2,657

1,902

234

326

266

159

101

308

1,476

1899

1262

1,844

1,148

3,240

1,836

185

120

288

174

326

1,252

1900

1251

1,951

1,191

3,731

1,948

262

153

308

184

362

1,457

1901

1246

1,962

1,199

4,134

2,044

326

210

325

197

101

386

1,641

1902

1213

1,949

1,197

4,421

2,087

429

219

840

217

403

1,800

1903

1196

1,926

1,185

4,605

2,101

516

173

364

238

438

1,917

1904

1164

1,888

1,177

4,666

2,115

652

127

387

265

101

425

2,054

1905

1162

1,912

1,190

4,813

2,212

520

214

402

286

117

428

2,065

1906

1161

2,106

1,274

5,199

2,344

421

155

413

306

104

460

1,959

The relatively small amounts spent on Dispute Benefit as
contrasted with the amounts on unemployment, super-
annuation, sickness and accidents are very noticeable.

During this period the funds in the possession of these
100 unions at the end of the year increased from £2,231,000
in 1897 by annual steps to £5,199,000 at the end of 1906 ;
even in this latter year they amounted to only about 2| times
the annual expenditure. Of this total £1,266,000 was held
by 16 unions connected with mining and quaiTying, and

STATISTICS OF THE WORKING CLASSES 165

£1,678,000- by 15 unions in the metal, engineering, and
shipbuilding trades.

3. The Friendly Societies have in the aggregate very much
larger funds and a much greater membership than Trade
Unions. The methods, objects and importance of the 29,000
societies registered vary so much that the gross totals show
very little, but are given to show the numerical importance
of these institutions. The 14 societies which are grouped
together in the following table have many branches and are
specially important.* The growth has been uninterrupted,
so that it is not necessary to give statistics for intermediate
years.

All Friendly Societies.

No. of registered societies
Membership .
Accumulated funds .

1897.

29,381

10,934,000

£35,736,000

1905.

28,954

14,596,000

£52,619,000

29,310

15,983,264

£57,128,168

Aggregate of 14 important Societies.

1897.

1905.

1907.

Adult male membership .
Yearly income ....
Total accumulated funds .
Total benefits of all kinds
Sick benefit . . . .
Funeral benefit

2,255,000

£3,969,000

£18,447,000

£2,479,000

£1,809,000

£383,000

2,642,000

£4,850,000

£26,408,000

£3,142,000

£2,290,000

£436,000

2,678,000

£5,129,000

£28,498,000

£3,436,000

£2,499,000

£466,000

The 14 societies own thus half of the total accumulated
funds shown for all the societies. The great part of the
expenditure is on Sick Benefit. It is not easy to deal with
the statistics of Friendly Societies in further detail in a
compact form, and the figures just given are rough. Con-
siderable detail is to be found in the Annual Reports of the
Chief Registrar of Friendly Societies.

* The four largest, viz. the Manchester Unity of Oddfellows, the
Ancient Order of Foresters, the I.O. Eechabites (Salford Unity), and
the Hearts of Oak have between them 1,870,000 members.

166 AN ELEMENTARY MANUAL OF STATISTICS

4 The Registrar of Friendly Societies also receives in-
formation as to Building Societies and as to Co-operative
Societies, which are summarized in the Annual Abstract of
LcAour Statistics. The former are not confined to the
workingrclass, and the statistics are not easy to interpret.
The latter hold a very important part of the aggregate of
working-class savings, and no small proportion of working-
class expenditure is accounted for in their statistics of sales.
The following table contains some summary statistics of these
societies :■ —

All Co-operative Societies in the
United Kingdom for which in-
formation is received.

1897.

1905.

1900.

Number of Members .
Capital, Share
Capital, Loan
Amount of Sales .

1,512,000

£18,099,000

£4,769,000

£59,684,000

2,185,000

£27,743,000

£8,386,000

£90,724,000

2,259,000

£28,835,000

£9,114,000

£93,964,000

Sales by Retail distribution
societies ....

Sales by Wholesale distribu-
tion societies .

Sales by other distribution
societies ....

£40,129,000
£16,442,000

£61,087,000

£27,891,000

£426,000

£63,354,000

£29,825,000

£676,000

For comparison with these figures it may be added that
the total paid in wages in the United Kingdom is estimated
roughly at about £700,000,000 per annum. Of course sales
are not exclusively to the working-class.

The statistics of the last three paragraphs suggest a very
interesting investigation, beyond the scope of the present
work, as to the aggregate savings of the working-class. Sir
E. W. Brabrook (then Chief Registrar of Friendly Societies)
devoted his Presidential Address to Section F of the British
Association in 1903 to this subject {Report, 1903, pp. 729-
740).

5; A great deal of attention has been given from time to
time in various countries to working-class " budgets," which

STATISTICS OF THE WORKING CLASSES 167

show the cost and amount of the various commodities on
which wages are spent. The information is always collected
first-hand from the workman or his wife, and it is not easy
to secure accurate accounts either of income or expenditure ;
for to include clothes and occasional earnings these accounts
should be spread over a long period, an undertaking that
requires intelligence, time and attention to minutiae on the
part of the informant. Often, in fact, the budgets do not
exactly balance ; expenditure on drink and luxuries tends to
be underestimated, and in the end the returns apply only to
specially thrifty households. So far as the items contained
in the following table are concerned these objections do not
apply. The tables are taken from the Reports on the Cost
of Living of the Working Class in the United Kingdom,
France and Germany (Cd. 3864, 4032 and 4512); these
volumes contain a great wealth of detailed information as to
wages, prices, rents and conditions in a large number of towns
in the countries named. The relative levels of wages, rents
and prices are compared by means of a system of index-
numbers which presents many statistical difficulties, and the
results based on them should be criticized closely in relation
to the data. As soon as we attempt to compare the well-
being of two groups of people we find that statistics of
incomes, prices and methods of expenditure only take us part
of the way ; habits, desires, thriftiness and skill in domestic
economy vary greatly from class to class and from nation to
nation, and cannot be reduced to statistical measurement;
but the data in these volumes make possible vivid descrip-
tions of the economic life of typical working-class families
throughout the towns of the three nations.

168 AN ELEMENTARY MANUAL OF STATISTICS

The figures given in the following tables, however, present
few statistical difficulties.

AvEBAGE Weekly Cost and Quantity op Food consumed by
Urban Workmen's Families, United Kingdom, 1904.

Limits of weekly income ,

Average weekly family iuconie
Average numl)er of children living at
home

Bread and flour

Meat (bought by weight) .

Other meat and fish .

Bacon

Fresh milk

Cheese

Butter ....

Potatoes ....

Other vegetables and fruit

Bice, tapioca and oatmeal

Sugar

Tea

Coffee and cocoa

Jam, etc

Other items

Total expenditure on food

Expenditure on bread and flour, as
% of food cost ....

Expenditure on flsh, meat and bacon,
as % of food cost ....

Expenditure on all food, as % of
income

Bread and flour .

Meat (bought by weight) .

Bacon . . . .

Cheese

Butter

Potatoes

Tea

Sugar

Fresh milk

1 4

14 4|

21
27

lbs.

28-4

4-4

•9

•7

1-1

14-0

•48

3 '9

pints.
5-5

30^. and I 35s. and
under 85s. under 40s.
31s. lljfi. 36s. 6id.

3-4

Cost.

.■!. d.

3 3j

3 4J

n-j

1 7
95
7
6
10
11}
Si

1 71

17 10}

19
27
66

s. d.

3 3i

4 3i
10
10}

1 3}

1 10}
lOJ
10
6

10}

1 Of
3J
6

20 9}

16
29
65

s. d.

3 4}

4 5^
1

1 4}
6

2
10}
ll|
6}

11}

1 1}
4}
6J

2 5

22 3^

15
29
61

Quantities.

40s. and

above.

62s. OW.

s. d.

4 3|

5 m

1 4

1 3}

1 43

1 7J

3 Oi

3 2}

lbs.

30-0

29-4

30 ■O

6-3

6-4

1-2

1-4

•7

■8

1-6

1-7

1^9

16-8

16-1

15^9

■65

■67

•69

4-6

4^8

6^2

pints.

7-7

9^8

lO'S

37-8

pints.
12-6

STATISTICS OF THE WORKING CLASSES 169
French Towns.

Limits of weekly income

Under
20s.

20s. and
under 25s.

25s. and
under 30s.

80s. and
under 35s.

36s. and
under 40.'(.

40s. and
above.

Average weekly family

income.

17s. gjd.

22s. lid.

278. 7|d.

32s. 4}rf.

37s. 33ti.

52s. lid.

Average number of chil-

dren living at home .

1-0

1-8

1-9

2-1

2-9

Cost.

s. d.

Bread

2 9

2 llj

8 H

3 4i

3 8

4 8

Potatoes ....

Meat, hacon, etc. .

3 2

4 ii

5 0|

a 2i

8 11

9 4

Sugar ....

Other items .

4 2J

5 8

6 Hi

8 Oi

9 1

12 3i

Total expenditure on food

U 1|

13 11

16 2}

18 8J

20 Hi

27 nj

Expenditure on bread, as

% of food coat

Expenditure on meat,

etc., as % of food cost .

Expenditure on all food.

as % of income .

Quantities.

lbs.

Bread

22-9

24-1

24-6

26-2

27-6

35-9

Potatoes ....

14-7

12-3

13-9

14-6

16-8

20-5

Meat

4-2

6-6

7-8

8-6

11-5

Sugar

1-2

1-6

1-5

1-7

1-8

2-2

German Towns.

Under

20s. and

25s. and

30s. and

35s. and

40s. and

Limits of weekly income

20s.

under 25s.

under 30s.

under 35s.

under 40s.

above.

Average weekly family

income.

17s. 7id.

22s. 8id.

27s. Id.

31s. lOJd.

368. 8d.

48s. 8id.

Average number of chil-

dren living at home .

2-4

2-3

2-5

2-6

2-8

3-8

Cost.

s. d.

Bread and flour

2 6J

2 7i

2 10

3 Oi

3 5

4 6

Potatoes ....

lOi

1 2*
8 11

Meat, bacon, etc. .

3 11

4 4J

6 8

Sugar ....

Other items .

4 7

6 4|

7 8

8 6i

9 7

12 1^

Total expenditure on food

12 H

14 Ih

16 lOi

IS lOi

21 11

27 4i

Expenditure on bread

and floiu-, as % of food

cost ....

Expenditure on meat.

etc., as % of food cost.

Expenditure on all food.

as X of income .

NTITIKS.

Qua

lbs.

Bread and flour

22-0

23-6

25-0

26-1

29-8

38-2

Potatoes .

28-1

25-6

24-0

23-8

24-6

33-5

Meat, bacon, etc. .

4-9

6-6

6-2

7-4

8-2

Sugar

1-8

1-S

2-0

2-1

170 AN ELEMENTARY MANUAL OF STATISTICS

It must be remembered that these tables do not show the
relative levels of wages in the three countries, but only the
distribution of expenditure of wages of given amounts. The
general agreement between the three tables is remarkable ;
in each country the proportion spent on food falls by about
one-sixth part as incomes increase from the lowest to the
highest ; the propo7'tion of the food expenditure devoted to
meat is nearly the same for all incomes, while the actual
expenditure increases. A difference is noticeable as to the
consumption of bread and flour ; in the United Kingdom it
is from 28'5 to 30 lbs. except for the large families shown in
the last column; in France and in Germany, on the other
hand, the lower the income the less bread consumed per head.
Among many interesting details we may notice the large
amount of sugar consumed in the United Kingdom.

CHAPTER IX

INCOME AND CAPITAL

1. By Total National Income is generally meant the aggre-
gate of the incomes (including earnings) of the persons com-
posing a nation ; income is taken as meaning the money, or
money value of goods, coming into a person's possession during
a year for his own use (subject to rates and taxes), after all
expenses connected with obtaining it are subtracted. The
earnings of the working-classes, discussed in Chapter VI are
thus measured, and incomes are assessed for income-tax on
the basis of this definition.

It is doubtful whether a perfectly definite meaning can be
attached to Total National Income. The sum of money
nominally representing it of course does not actually exist ;
a great pa,rt of income is actually received in the form of
cheques which are exchanged for services, and the total is
more correctly the total estimated value of services rendered
to, or commodities consumed by, the members of the nation,
together with the addition to savings, that is to capital goods.
In such a total are included the services of an agricultural
labourer at £3 per month and of a physician at the same
price for a short visit, the value of a day's sojourn at a hotel
and the equal value of 60 quartern loaves of bread or 80 oz.
of tobacco. It is doubtful whether the same unit, £1 sterling,
can in any real sense be used to measure such diverse and
non-interchangeable services and commodities. This line of
argument would lead too far from our subject, but it is im-
portant as emphasizing the fact that the hundreds of millions
of pounds which make the aggregate are not a homogeneous
total and cannot be used for processes of averaging without
analysis. To say that the average income of the inhabitants

171

172 AN ELEMENTARY MANUAL OF STATISTICS

of the United Kingdom is £40 is nearly meaningless, except
as an arithmetical entity for use in arithmetical processes.
The total depends on the existing method, and the mo-
mentarily resulting scale, of valuing various services and
commodities; the scale is continually changing, and the
total would easily be affected, for example, by a redistribu-
tion of income by taxation or under a socialistic regime.

Nevertheless, the total and resulting averages can be used
for comparing total or average income or wages through a
period so short that during it no great changes in valuation
or in distj'ibution have taken place.

2. The aggregate of the earnings of the wage-earning class
is generally estimated by calculating the average annual
earnings of men, women and children from the statistics
described in Chapter VI, and multiplying these averages
by the numbers of persons occupied, as indicated by the
census. There is much that is hazardous in this method,
but it seems probable that the aggregate of net earnings
received by persons working for hire (including a valuation
for payments in kind, etc.), is in round numbers about
£700,000,000 annually in the United Kingdom. This, of
course, ignores completely the value of the unpaid domestic
work done by women for themselves or their families or
relations, and of many other unpaid services.

The aggregate of the incomes of those whose income is
over £160 per annum is estimated from the income-tax
returns at between £800,000,000 and £900,000,000, as will
be presently shown. This total includes earned and un-
earned income. A very small fraction of this has already
been included under the former total of earnings, for some
workmen pay income-tax.

Besides these two sums there are the incomes of those
who neither work for wages nor receive as much as £160
annually as incomes. No reasonably accurate method has
yet been devised for measuring this sum, but it is guessed
on the basis of rough estimates to be well over £200,000,000.

The aggregate of incomes of all kinds is thus estimated at

INCOME AND CAPITAL

173

about £1,800,000,000 in the year 1908, but this total must
be regarded as subject to considerable error, perhaps as much
as 10%.

Estimates on a similar basis for previous years show —

Population of

United Kingdom.

£000,000's

00,000's

1860

700

28,8

1870

950

31,3

1880

. 1,150

34,6

1890

1,350

37,5

1900

. 1,650

41,2

1908

. 1,800

44,5

These numbers are rough and uncertain, but they are
better than no estimates, and can be used for such purposes
as comparing the burden of taxation at different periods.

It must be remembered that the purchasing power of
money diminished between 1860 and 1874 (see Chapter IV),
increased till 1895, and fell again till 1907.
, 3. The statistical tables in the Annual Reports of the
Inland Eevenue Commissioners are full of pitfalls even for
the wary. Their general nature can be best shown by the
report for the Fiscal Year ending April 1, 1907.*

■g X

Sff,

s«

§£

1906-r.

09 u

§1
1=3

ij ft

ill

o g o

£.

Schedule

00,000's

263,7

31,6

10,3

1.1

7.0

39,0

—

16,3

106,3

158,5

17,4

11,3

1,*

—

13,3

4.1

46,7

1.2

2,2

—

6,0

41,7

518,7

10,0

66,7

6,2

S,i

—

17,1

36,2

137,6

381,0

97,1

34,9

2,8

—

4,3

42,4

64,7

Totals

943,7

54,9

114,6

9,2

11,1

39,0

17,1

58,2

308,7

640,0

* Cd. 4226. For the year 1907-8, Cd. 4868, the statistics were not
completed for the Annual Report.

174 AN ELEMENTARY MANUAL OF STATISTICS

Schedule A includes profits from the ownership of lands
(£52,000.000), and houses (£210,000,000). For the former
one-eighth, for the latter one-sixth, of the profits have been
excused from taxation since 1894 for repairs.

Schedule B consists of profits from the occupation of land ;
these are simply assumed to he for purposes of taxation one-
third of the annual value of the farms, etc. The actual profits
are not stated at all except by a very small number of farmers
who elected to be assessed under Schedule D. The great part
is exempted or abated.

Schedule C consists of profits from British (£15,000,000),
Indian (£9,000,000), Colonial (£14,000,000), and Foreign
Government (£8,000,000) securities.

Schedule D contains profits from "Businesses, Concerns,
Professions, Employments (except those contained in Schedule
E), and certain interest." Railways in the United Kingdom
(£42,000,000), Mines (£16,000,000), Gasworks, Ironworks,
Waterworks and Canals, Foreign and Colonial Securities
and Businesses (£49,000,000), Loans on the Public Rates
(£7,000,000), and some other small items are separately dis-
tinguished; but the great bulk of the Income in this
Schedule (£373,000,000 out of £519,000,000 gross) is
lumped together. The deduction allowed for wear and
tear of machinery has increased considerably in recent
years.

Schedule E is entirely composed of the salaries of Govern-
ment officials (£24,000,000), and officials of corporations and
public companies (£74,000,000).

4. Incomes, which are under £160 from all sources taken
together, are exempt from taxation. The total (£54,500,000)
in the column in bhe above table shows only the sum that
accidentally came to the notice of the Commissioners, some
of which was charged with income-tax collected at the source ;
on this in 1906-7 £846,000 was repaid, while on the remainder
no tax at all was paid. This total merely shows that as much
as £54,500,000 is received annually by persons whose incomes
are below £160, and of this £33,000,000 under Schedules A

INCOME AND CAPITAL

175

and C is from property ; but it evidently does not show the
aggregate of such incomes, for no tax is paid on and no
notification is made to the commissioners as to the majority
of small salaries.

Abatements are made on all incomes proved not to exceed
£700 ; that is, a certain sum, varying from £160 for incomes
not exceeding £400 to £70 for incomes exceeding £600 and
not exceeding £700, is deducted by the surveyor of taxes
from the total income, and tax is charged only on the
remainder. To obtain an abatement it is necessary to state
the amount of total income, as defined by the Commissioners,
from all sources. For the year 1906-7 the following claims
were allowed : —

Range of incomes.

Amount of abate-
ment on each income.

No. of abatements.

Total amount of
abatements.

£
160-400
400-500
500-600
600-700

£
160
150
120

628,818
58,704
33,150
16,607

£000-s

101,610

8,806

3,978

1,162

737,279

£114,557

The number of abatements allowed during 10 years have
been ; —

Year.

Total No

of abatements.

Full rate of tax.

OOO's

s. d.

1898-1899

543

1899-1900

577

1900-1901

601

1901-1902

636

1 2

1902-1903

664

1 3

1903-1904

696

1904-1905

709

1905-1906

725

1906-1907

737

1907-1908

About 764

176 AN ELEMENTARY MANUAL OF STATISTICS

The rapidity of this increase is supposed to show that,
as the income-tax has been increased, fewer persons have
neglected to take the necessary trouble to substantiate their
claim ; and it is supposed that there are still many persons
with incomes between £160 and £700 who do not appear in
this table. The total number of persons with such incomes
was estimated * to be about 840,000 in 1904-5, of whom only
709,000 claimed abatement.

Allowance is made for Life Insurance Premiums, not ex-
ceeding one-sixth of the net personal income, paid by the tax-
payer on his life or on his wife's life.

Charities, Hospitals, Friendly Societies and some other
institutions are relieved of income-tax.

5. The statistics of Gross Income are often quoted as show-
ing the growth of income as a whole, but they include much
that is not income, and the allowances have increased con-
siderably in recent years. The best plan of statement seems
to be as follows : —

Income on whioh tax was
received and not refunded .

Abatements on incomes from
£160 to £700

Insurance payments

Charities, hospitals, etc.

Repairs .

Wear and tear .

Other allowances

£764,000,000 personal f in-
come above £160.

Not personal income.
Not income.
Mostly not income.

t This includes, however, incomes of clubs and similar bodies and a small amount of
income of municipalities and other local authorities.

Working through the income-tax reports, and allowing for
the deductions on the present basis as accurately as possible
in earlier years, we have the following table : —

* Report of Committee on Income Tax (H. of C, 365 of 1906),
229.

INCOME -AND CAPITAL

177

Estimated Aggregate Personal Income above Exemption Limit.

[Bxemption limit, 1860-1876, £100 ; 1877-1893, £150 ; 1894-1907, £160.]

Fiscal year.

£0DO,O00's

£000,000's

1859-1860

254

1884-1885

505

1860-1861

254

1885-1886

498

1861-1862

244

1886-1887

496

1862-1863

273

1887-1888

504

1863-1864

285

1888-1889

518

1864-1865

309

1889-1890

544

1865-1866

326

1890-1891

568

1866-1867

335

1891-1892

569

1867-1868

341

1892-1893

572

1868-1869

344

1893-1894

562

1869-1870

355

1894-1895

553

1870-1871

385

1895-1896

567

1871-1872

399

1896-1897

587

1872-^873

430

1897-1898

611

1873-1874

461

1898-1899

641

1874-1875

482

1899-1900

663

1875-1876

490

1900-1901

695

1876-1877

473

1901-1902

715

1877-1878

476

1902-1903

720

1878-1879

470

1903-1904

732

1879-1880

462

1904-1905

738

1880-1881

468

1905-1906

753

1881-1882

481

1906-1907

764

1882-1883

493

1907-1908

796*

1883-1884

507

* Estimated from incomplete data.

Average income has risen about as fast as the exemption
limit during the whole period. The number of income-tax
payers is not, and cannot be, known directly from the report ;
it was estimated f at about 1,000,000 in 1906. The table
just given probably shows the general features of the growth
of that part of the national income which is subject to income-
tax with fair accuracy, and the rate of growth may accurately
be deduced over quite short periods, if no exceptional event
occurred in them ; but there are many difficulties, some still

•j- The details arising from the report are so troubleaome to handle
that this estimate is not discussed here.

178 AN ELEMENTARY MANUAL OP STATISTICS

the subject of controversy, in such an estimate, which we will
enumerate without discussion : —

The amount shown for a year (say 1906-7) is the total
income in respect of which the tax was paid or remitted in
the year ending April 5 (1907). Under Schedule D more than
half (£373,000,000 profits on businesses not otherwise detailed)
is assessed on the average profi.ts of the preceding three
years (presumably 1903, 1904, 1905), mines (£16,000,000)
are assessed on the average of the preceding five years, and
about £54,000,000 more on the profits on the preceding year.
The whole assessment for 1906-7 may be regarded as relating
to a short period whose centre is the Calendar year 1905 ;
the whole table should be set back, therefore, about a year,
and the peculiarities of individual years are averaged away.
Thus the high profits in 1907 will continue to have effect
on the figures till the year 1912-13, for which the Report
will presumably be published in the autumn of 1914 !

From the 52nd Report (for the year 1908-9) onwards, a
new regulation affecting the assessment of former profits will
affect the figures; this prevents the statistics for 1907-8
being yet completed, and a resulting alteration of the form
of the returns is likely to destroy the comparability of the
statistics from 1908 onwards with earlier ones. In this 52nd
Report, p. 138, means are given of equating the old and new
methods in 1907-8.

It is generally supposed that greater vigilance and new
powers of the surveyors of taxes have disclosed from 1907
onwards considerable amounts of income, which had hitherto
evaded taxation. If this is so, the amounts for years prior to
1907 should be somewhat raised for comparison with 1907
and later years.

It is believed that some part of the income which is
received from abroad, and is liable to taxation, successfully
evades taxation ; natu rally this amount can only be guessed.
It is not improbable that in recent years the net of the
commissioners has become finer and wider, and that less and
less escapes. Actually £80,000,000 paying tax was identified

INCOME AND CAPITAL 179

in 1906-7 as income from abroad, and besides this there are
other large sums included in Schedule D (52nd Report,
pp. 163-5). If less escape than in former times, earlier
figures should again be increased for comparison with more
recent.

To get the total income above £160 it would be necessary
to add an estimate for such income from abroad as escapes,
and also an estimate for profits of trades and professions
which are generally believed to be on the whole under-
valued. £80,000,000 is a current guess for these two
amounts together, but in fact there are practically no data
for an estimate.

6. Recently the gross returns for income-tax have been
placed alongside the returns of Changes of Wages, discussed
in Chapter VI above, and the conclusion drawn that income
has grown while wages have been nearly stationary from
1900-8. It is possible that this may be true, but the
relative rates of growth cannot be shown from the statistics,
for the following reasons : —

The wage-changes published only apply to a small part ot
the working population, and afford no test of the general
growth of wages (pp. 141 seq, above).

The most recent statistics available are for the income
assessed * for 1907-8 (and even these are incomplete), and
these belong to 1906 rather than to any other year.

The income-tax returns cannot be allotted to an individual
year.

The relation of gross to net income has changed.

The collection of the tax has recently been more
thorough.

The total net income, as shown in the table above (p. 177),
naturally grows 1 % per annum with population, while the
wage-changes have no relation to population.

* It is true that the net rec&ipt in the year 1908-9 is known, and is
6 % higher than that for 1907-8 ; but the net receipt in a year and the
net produce for the year are not the same, for considerable sums are
paid^more tlian a year late. (52nd Report, pp. 136 and 138.)

N 2

180 AN ELEMENTARY MANUAL OF STATISTICS

The year 1900, which is frequently taken for comparison,
was a year of exceptional inflation for wages, but is a normal
year in the income-tax returns.

The following table shows the present writer's estimate of
the change of average wages, and of the average income of
the income-tax payer, each expressed as a percentage of the
level in 1880. The former is from p. 147 above, the latter is
based principally on the method of the table * on p. 177, with
the years adjusted and allowance made for the growth of
population, and with some other modifications based on the
discussion in paragraph 5 above.

Index-numbees op Incomes and Wages, f

Wages.

Incomes.

Wages.

Incomes.

Wages.

Incomes.

1880

100

1890

114

105

1900

130

117

1881

100

1891

115

103

1901

128

117

1882

103

1892

115

100

1902

126

117

1883

103

101

1893

115

100

1903

125

117

1884

103

100

1894

115

101

1904

123

118

1885

101

1895

115

103

1905

123

119

1886

100

1896

115

107

1906

126

123

1887

101

1897

116

109

1907

133

—

1888

104

103

: 1898

120

111

1889

110

107

1899

123

113

t There is no allowance for the cycle of unemployment in this table. Such allowance
would raise the numbers in some years and lower them in others, without affecting the
general run. The averaging of the incomes under Schedule D also merges together good
and bad years for the income Index-number.

In making this computation care has been taken to exclude
the same proportion of income as exempt (see Uconomic
Journal, 1904, p. 460), so that the intermediate class who are
not wage-earners but have small incomes are excluded from
the calculation on the same proportionate basis throughout.

It is probable that the complete figures for 1907 will
show a rise in incomes, and those for 1908 a slight fall in
wages.

* See Economic Journal, 1904, p. 45!).

INCOME AND CAPITAL 181

7. Material is accumulating for the eitimation of income
earned as opposed to unearned (or derived from capital).
From the year 1907-8 any individual whose total income
did not exceed £2,000 is entitled to such relief as would
reduce the rate on his earned income to 9d. in the £, abate-
ments for income under £700 being first deducted from
earned income. For the year 1907-8 relief was granted in
approximately 750,000 cases, and the amount of income
taxable at Qd. was estimated at £187,700,000, subject to
abatements for incomes under £700 of £87,000,000, and for
Life Insurance Premiums of £6,000,000 (52nd Report, pp.
138-9). The number is somewhat greater and the income
somewhat less than was expected from the estimate made for
the Committee on the Income Tax (1906), where the total
earned income was estimated to be £320,000,000 ; the number
of persons earning heivieen £160 and £2,000 was estimated as
628,000, and the number receiving between £160 and £2,000
from all sources was estimated as 900,000.* If all these figures
are correct, it appears that a large number of income-earners
have also unearned income which brings their total over
£2,000, or else did not claim the lower rate, so that much
earned income paid tax.t It is known (52nd Report, pp.
172-3) that there is a sufficiently large number of persons
earning less than £160, whose unearned income brings them
above the exemption limit, to account for the difference in the
numbers.

No doubt these estimates will be carefully examined by
statisticians in the near future.

8. The " unearned " income in 1906-7 may be set out as
follows : —

* Report of Committee, pp. 220-229 ; evidence by the present author.
All the figures were for 1903. On the same basis the earned income for
1906-7 would be £330,000,000.

f We have only statistics for the one year, in which the notice given
was short and claims were not admitted after September 30. There are
also various possibilities of misinterpretation of the published informa-
tion.

182 AN ELEMENTARY MANUAL OF STATISTICS

Nat income, together
with abatements and
insurance.

Schedule A

Schedule C

Schedule D

fOOO.OOO's

170

239

Total received by persons with more than £160 .
Exemptions under £160

453
43

Total unearned income known to the Commis-
sioners ... ....

£496

Besides this are probably some additions (as detailed in
paragraph 5 above) for income escaping tax, and a small sum
accruing to persons with less than £160 per annum which is
not reviewed by the Commissioners. On the other hand,
some part of the income shown belongs to societies (clubs,
colleges, etc.), and is not personal; Sir H. Primrose estimated
this in 1906 at £50,000,000.

9. The aggregate capital owned by the individuals of the
nation can be estimated either by capitalizing the " unearned "
income, or from the records of estates paying death duties.
The first method was used by Sir R. Giffen in his essay on
" Recent accumulations of capital in the United Kingdom," *
1878. The latter has been the subject of much recent work.
Unfortunately, it is extremely difficult to reconcile the results
reached by the two methods.

To use the records of estates, assessed for Estate Duty, it
is necessary to estimate the number of estates in existence
in relation to the number which pass per annum. The best
estimate appears to be that by Mr. B. Mallet in the Statistical
JouTTial, March 1908, where by tabulating the values of the
estates according to the age of the deceased, and multiplying
by the reciprocal of the death-rate age by age, he arrives at
the multiplier 24 ; that is, he concludes that 24 times the

* Essays in Finance. Also in Statistical Journal, 1878.

INCOME AND CAPITAL

183

value of estates passing in one year gives the total value of
such estates in existence. A higher multiplier had been
used in previous estimates, but this neglected the important
fact that estates as a whole increase with the age of their
possessors.

The following table shows the results of this estimate in
relation to the income-tax returns. It is modified from
that on p. 220 of the Report of the Committee on the
Income Tax.

United Kingdom.

AsBessed value of estates

reviewed for estate duty.

Average of 10 years,

1894-1904, multiplied by 1-1

to bring up to 1904-5.

Presumed
assessed

value of all
estates,
24 times
previous
column.

Corresponding in-
come, 1904-6, from
income-tax returns.
[Allowances de-
ducted from gross
income, but not
insurance, abate-
ments or exemp-
tions.]

Averages deduced from
previous columns.

Bate of
Interest
per cent.

Number
of years'
purchase.

Millions
Stoclcs, companies, mort-
gages, bonds, mines
and quarries . . £127
Agricultural land, tim-
ber, building land . 26
Houses, and all rents
that can possibly be
connected therewitli . 03

Millions

£3,050
625

1,610

Millions

Companies,etc.£265
Lands . . 41

Buildings. . 163

8-7
6-6

10-0

114
15

£216

Goodwill, share in Arms,
book debts, stock-in-
trade, half cash at
bank . . . . 29

5,185
700

£459
Unknown.

No corresponding
income.

8-9

£246

Insurance, debts, small
sundry properties, per-
sonal goods, half cash
at bank ... 35

5,885
840

£280 £6,726

It is evident that the rates of interest shown in this table
are higher than those in fact obtained. Indeed, in the
Report of the Commissioners (Cd. 2663),* the net income
from lands is stated at 4'3% (instead of 6'6^), and from

* See also the table, pp. 80-1, in the 52nd Report (Cd. 4226).

184 AN ELEMENTARY MANUAL OF STATISTICS

buildings at 5-5% (instead of 10%). Either, then, (i) the
multiplier 24 is too low, or (ii) estates are undervalued for
probate, or (iii) very considerable sums pass inter vims and
do not come up for probate, or (iv) the income-tax returns
contain income on property that is not subject to probate, or
(v) some part of the sum for companies, etc., above should
be transferred to " earned income." It does not seem possible,
even when all these considerations are given full weight, to
bring the estates up to the income, and till this reconcilia-
tion is effected the total national capital cannot be safely
estimated.

10. It is probable, however, that the increase of the total
value of estates liable to duty observed over a period long
enough to eliminate the accidents of individual years has a
close relation to the growth of capital. The following table
shows these values since the commencement of the duty.

Net capital value of estates which become
liable to estate duty.

No. of millionaires
included.

1895-96

£000,000's

213

1896-97

219

1897-98

247

1898-99

251

1899-00

293

1900-01

265

1901-02

289

1902-03

270

1903-04

264

1904-06

265

1905-06

272

1906-07

298

1907-08

282

1908-09

271

In the first two years the totals are those of the capital on
which duty was jpaid, which is less than the capital liable to
duty which is that shown for the other years, since the
payment is in some cases made in instalments.

CHAPTER X

TAXES AND BATES

1. The following table shows the "Amount of the Imperial
Eevenue (Exchequer Eeceipts, net) of the United Kingdom
adjusted by deduction of charges in the year ended March
31,1909."*

O.OOO'S 0,000's

Ciistotna . . . .

£29,20

Inland Bevemie —

Excise

£33,65

Estate, etc., duties

18,37

Stamps (exclusive of Fee and

Patent Stamps) .

7,77

Land Tax . . . .

House duty ....

1,90

Property and Income tax

33,93

T'otal TnlflTirl T?pvpmip

Ofi 9n T'ntTl rrvmnr

■L LfLn^ -l-UlclIllA XliCVCllLlC .

frnm tiTm Pl^^i 'I'n

Post Office and Telegraph Ser-

vice —

Net receipts .

. . . . 4,81

Crown Lands ....

. 53

Total from

Suez Gcmal Shares .

. 1,06 property . 1,59

Total net revenue .

. £131,95 £131,95

* Detail, sufficient for most purposes, of revenue and expenditure is
given in the Statisiiical Abstract ; the Annual Financial Accounts of the
United Kingdom {e.g. H. of C, 200 of 1909, price 5|d.) contains the
more detailed official statement ; the reports referred to in Chapter IX
above give details of the Inland Revenue. The Commissioners of the
Customs also issue a small Annual Report.

185

186 AN ELEMENTARY MANUAL OF STATISTICS

In the Exchequer receipts, unadjusted, there are also
included £113,000 from interest or repayment of Sundry
Loans (of which those to the Cunard Steamship Company
and to the Gold Coast Colony account for £91,000), mis-
cellaneous receipts amounting to £1,004,000, and receipts for
Fee and Patent Stamps amounting to £1,023,000. These
sums are received by, and redistributed from, the Treasury,
but are treated as of the nature of cross-accounts, and do not
enter into the balance-sheet of taxes and expenditure as
generally understood.

2. The corresponding " Amount of the Expenditure (Ex-
chequer Issues, net) of the United Kingdom, adjusted by
deduction of extra receipts in the year ended March 31,
1909," is shown as follows : —

0,000's

I. National Debt services . £31,39

II. Naval and military expenditure 56,78

III. Civil Services —

(1) Civil List and Civil Administration —

(a) Charged on Consolidated Fund . £1,38
(6) Voted . . ... 10,69

Less extra receipts . . . 1,76

Net 10,31

(2) Elementary education 15,44

(3) Old Age Pensions 2,07

(4) Charges transferred from local to imperial funds —

(o) Charged on Consolidated Fund . 9,82

(b) Voted 3,59

Less extra receipts .... 6

Net . . ... 13,35

IV. Customs and Inland Revenue —

Customs .... ... 97

Inland Revenvie ...... 2,35

3,32

Total net Expenditure .... £132,66
Total net Revenue 131,95

Excess of Expenditure over Revenue . 71

As to I. The National Debt Services include a permanent
annual charge applied to Interest, £19,000,000, and a sum

TAXES AND RATES

187

for repayment or reduction of debt (£9,000,000), and various
smaller charges connected with temporary borrowings under
various Acts.

As to II. The Naval and Military Expenditure is the total
actually spent in the year, less sums charged to the National
Debt. The account was —

1907-8

1908-9

Anny expenditure . ...
Navy expenditure

£00,000's

27,3
31,1

£00,000's

26,8
32,2

Charges not met in the current year, but
transferred, mainly to National Debt
Services ....

58,4
2,3

59,0

2,2

Net expenditure .

£56,1

£56,8

There are also smaller items which complicate the account.

As to III. (1 a) The charge on the Consolidated Fund
(£1,380,000) consists of the Civil List (charges for the ex-
penses of the Crown), Annuities, Pensions, Salaries of Judges
and other officials.

(1 b) The vote is for the annual expenses of the Adminis-
trative and Executive departments of the Civil Services.

(2) The vote for Elementary Education corresponds to the
expenses of the Board of Education (£13,600,000), Public
Education in Scotland (£2,000,000) and Ireland (£1,500,000),
less payments in respect of Secondary Education. The
residue, added to the sum paid from the rates (see p. 194
below) is the total public expense of elementary education,
including administration.

(3) This payment for old age pensions is for only a fraction

of the year.

(4) Under this heading come the sums collected as taxes
and used in relief of rates, or (what is nearly the same thing)
for local purposes which would otherwise be paid for out

188 AN ELEMENTARY MANUAL OF STATISTICS

of rates, (a) The principal items here are £1,300,000
"Additional Beer and Spirit Duties" allotted since 1891 to
Secondary or Higher Education, £4,400,000 allotted from
the Estate, etc.. Duties to the relief of ratbs generally and
of agricultural rates in particular, and £3,000,000 received for
licences and handed to the local authorities. (&) The votes
account for many classes of payments, of which the largest
items are : about £600,000 in lieu of rates on Government
Property, about £1,100,000 for Prisons and Reformatories,
and about £1,400,000 for the Irish Constabulary and Dublin
Police.

As to IV. This heading covers the expense of collecting the
taxes, and is a remarkably small percentage of their yield.

The details of, and relation between, the unadjusted and
adjusted Revenue and Expenditure, of which only the latter
are given here, can be studied in the Finance Accounts and
in an explanatory memorandum called " Public Income and
Expenditure " (e.g. H. of C, 258 of 1908, price 5d.).

Corresponding figures for recent years are given on p. 189.

3. Turning to the details of Revenue, we find the receipts
under Customs and Excise were as follows : —

1894-5.

1902-3.

1908-9.

Customs —

£0,000's

JEO.OOO'S

Spirits .....

Sugar

Tea

Tobacco

4,.39

3,59
10,42

4,96

4,48

6,00

12,45

3,96

3,16

6,05

13,82

Wine

Exported coal ....
Imported corn and grain
Others

1,14
80

1,52

1,99

2,35

1,12
1,05

Excise —

Beer

Spirits

Licenses . . . .

Others . ...

10,49

16,00

3,75

13,71

19,03

4,26

12,69

17,46

3,11

Total, Customs and Excise .

50,85

72,14

62,85

TAXES AND RATES

189

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190 AN ELEMENTARY MANUAL OF STATISTICS

The coal export duty was in force for six years, 1901-7 ; that
on corn a little over a year ; the sugar duty was introduced
in 1901. In 1902-3 the total receipts were at their
maximum for the period. In 1908-9 the various licences
in respect of the manufacture and sale of alcoholic drinks
amounted to about £2,200,000.

The estate duties, etc., were referred to in the preceding
chapter. The principal items are Legacy Duty, Succession
Duty, and Estate Duty. The last was introduced in 1894,
in substitution for, and increase of, former duties, but the full
yield was not realized immediately, while the year 1894-5
contains part of the old duties.

189t-5.

1896-7.

1902-3.

1908-9.

Legacy duty
Succession duty .
Estate duty .
Miscellaneous

£0,000's

2,81
1,35
3,81
2,92

£0,000's

2,55

10,23

iO.OOO's

3,00

13,82

£0,000's

3,91

14,36

10,89

13,88

17,91

19,11

The receipts from Stamp Duties, Land Tax and Inhabited
House Duty have varied little during the fifteen years under
consideration.

The Property and Income Tax was discussed in the last
chapter. The tax was at *Id. in the £ in 1893-4, Sd.
1894-99, Is. in 1900-1, Is. M. in 1901-2, Is. M. in
1902-3, l\d. in 1903-4, Is. in 1904-9, and Is. 2d. in
1909-10. Exemptions and abatements were extended
in 1894 and 1898.

4. The Inhabited House Duty is specially interesting for
the statistician, for in the tables relating to it (52nd Report
of the Commissioners of the Inland Revenue, pp. 113 sqq^ we
have information as to the assessed value of all the inhabited
houses and residential shops and premises, and in less detail
of uninhabited premises, in England, Wales and Scotland.

TAXES AND RATES

191

The duty is not imposed ia Ireland. The following tables
show the nature of the information : —

Great Britain, 1907-8.*

Exempt from duty.

No. of premises.

Annual value.

£000' s

Premises not used as dwellings

664,266

49,819

t Separate dwellings exempt

from duty ....

64,681

845

Koyal and diplomatic residences,

hospitals, schools, etc. .

33,872

4,089

Houses of annual value —

under £10 .

3,162,752 ]

20,130 )

£10 and under 15

1,985,639

> 6,112,736

23,463 } 59,966

15 and under 20

964,345

16,373 )

Private

Charged to duty.

dwelling-

Others, t

dwelling-

Others. J

houses.

itOOO's

£000's

f'Separate dwellings"—

£20 and under £41 .

19,261

—

486

—

41 „ 61 .

4,695

—

234

—

Houses —

£20 and under £25 .

369,640

85,219

8,069

1,820

25 ,

30 .

248,531

65,183

6,595

1,708

, 41 .

402,454

123,662

13,818

4,248

50 .

103,352

34,115

4,600

1,532

61 .

123,072

50,144

6,646

2,735

80 .

61,151

29,821

4,195

2,079

100 .

38,245

20,919

3,300

1,812

100

150 .

44,581

22,435

5,227

2,625

150

, 200 .

16,468

9,154

2,733

1,514

200

300 .

13,460

7,138

3,137

1,670

300

400 .

5,199

3,122

1,725

1,040

400 ,

.500 .

2,370

1,531

1,024

664

500

1000 .

2,826

2,328

1,827

1,507

1000 and over .

970

836

2,093

1,870

1,4.56,275

455,607

£65,710

£26,825

• The statistics are suhjeot to slight additions when anears are collected.

t That is, parts of buildings (e. g. flats) used as separate dwellings.

J Residential shops, hotels, public-houses, etc., farmhouses, lodging-houses.

192 AJSr ELEMENTARY MANUAL OF STATISTICS

In a different form the statistics show :-

Nos. of private dwelling-houses, whether charged

to or exempt from duty.

Metropolis.

Rest of
England.

Scotland.

Great
Britain.

OOO's

"Separate dwellings" exempt

—

„ „ £20 to £61

—

Houses to £10 .

2,580

575

3,163

£10,, 15 .

1,765

178

1,986

15 „ 20 .

803

964

20 „ 25

248

370

25 „ 30

169

249

30 „ 41 .

125

243

402

41 „ 100 .

108

188

3'26

100,, 500

500 .. .

—

608

6,060

965

7,634

The importance of these statistics is in their relation to
the social grading of the people, a subject with which the
population census does not deal, and also in relation to the
statistics of income. The income-tax returns and the value
of houses cannot easily be compared, but there is here a
possible field of investigation of a difficult character. In
general (but with many exceptions) there is one private
dwelling-house to one payer of income-tax, and also in
general (but with some extraordinary exceptions) the higher
the income the larger the value of the house occupied, but
the smaller the proportion of income spent on rent; this
proportion probably varies from 25% for some classes of
workmen in the large towns to 10% for persons with an
income of £700 a year. The number of income-tax payers
in Great Britain is probably a little ]ess than the aggregate
number of houses of value above £30 in London and above
£25 in the rest of Great Britain. The aggregate annual
value of these houses is £55,000,000 ; the aggregate income
of income-tax payers in Great Britain is somewhat over
£600,000,000. There is much that is hypothetical in this
comparison, but it suggests an interesting line of analysis. An

TAXES AND RATES

193

inquiry by sample as to the average income of the inhabit-
ants of different types of houses would lead to an independent
estimate of the aggregate of individual incomes in Great
Britain.

5. From the table of expenditure on p. 186 above, and the
explanation under III, p. 187, it is clear that Local and
Central Expenditure cannot be separated from each other;
and though rates and taxes are generally paid to different
authorities, they are equally a compulsory drain on the pockets
of the payer. We will, therefore, investigate the total sum
expended locally in the year 1905-6, the last for which the
figures are complete. The tables in the Statistical Abstract
on Local Finance need careful interpretation.

Local Authorities.

Heceipts, 1906-6.

England

and
Wales.

Scotland.

Ireland.

United
Kingdom.

£00,000's

iOO.OOO's

£00,000'3

£00,000'»

Eates

58,.3

5,7

3,1

67,1

Government contributions .

19,8

2,3

1,3

23,4

Tolls, dues, and duties

4,4

1,2

6,0

Bents, interest, sales of pro-

perty ....

3,3

4,0

Fees, fines, etc. ,

1,2

1,4

Water, gas, electric light,

tramways, and light rail-

ways ....

20,3

4,3

24,7

From loans .

24,5

3,3

2,0

29,8

Miscellaneous

4,6

5,8

Repayments for improve-

ments ....

1,8

—

1,8

£138,2

£18,0

£7,8

£164,0

The rates by themselves in 1906-7 amounted to £68,600,000
for the United Kingdom.

It is not practicable without a long investigation to allot
the whole of this £164,000,000 to categories of expenditure,
and there are innumerable cross-accounts with the Central
Government, with Capital and Interest balances, with muni-
cipal trading undertakings, and with the allotment of
particular receipts to particular purposes. The table is given

194 AN ELEMENTARY MANUAL OF STATISTICS

mainly for the purpose of exhibiting the difference between
the sum drawn in rates and total receipts.

6. It is not at all easy to estimate the annual expenditure
in direct and indirect taxation and in rates, the receipts of
which are applied to public purposes, national or local,
and to separate it from receipts from and payments for loans,
interests and profits. The following table, however, gives a

Public Kbceipts and Bxpendituee.

1894-6.

1905-6.

1900-r.

National eeceipts.
Indirect taxation —

Customs, excise and stamps .
Direct taxation—
House duty, land tax, income tax,
and estate duties

£000,000'8

57
29

£000,000'8

78-5
5i-5

£000,000's

77
53

Total from taxes .
Profits from post office *
Crown lands and Suez Canal

86
3

130
5-5
1-5

130
5-3

1-5

Total national receipts
Local receipts.
Bates

90
40

137
67

137
69

Total receipts

130

204

206

Expenditure —

National debtf, army, navy, civil list,
and civil services |, excluding
education

Education § (including administra-
tion), wliether paid from taxes or
rates

Poor relief |

Other local purposes met directly
out of rates and taxes .

18
12

107

106

tioo

Total expenditure .

130

204

206

* This may be regarded either as indirect taxation or as trading profits.

t Interest and repaj'ment, allotting the excess of income over expenditure shown on
p. 189 above to repayment of debt.

X Including expenses of customs and inland revenue.

§ Of course, large sums are privately subscribed and spent on education, as well as the
amounts here given.

II "Expenditurebyunionsandparishesinrelief of thepoor" ; expenditure, not outof loan,
in relief of the poor and cost of public lunatic asylums, in 1005-6, was about £18,000,000.

TAXES AND RATES 196

rough idea of the public contributions and the main objects
of expenditure. The statistics for 1906-7 cannot yet be
completed.

The figures in the line " Other Local Purposes " are ob-
tained simply by subtraction of the expenditure detailed in
the previous lines from the total receipts.

From the data given on p. 172 above, it appears that as
much as ^th or -Jji-th of the Aggregate of Individual Incomes is
taken in rates and taxes, direct or indirect.

From 1894 to 1905 total income probably increased about
30 %, rates and taxes together about 60 %. A very rough idea
of the change since 1860 may be given as follows. Kates
and taxes as a percentage of total income : — 1860, 12 %;
1880, 9 % ; 1890, 8 % ; 1894, 9 % ; 1905, 11 %. It is certainly
probable that the percentage fell from 1860 to 1890, and the
rough figures indicate that the grant of old age pensions
(which was, of course, subsequent to 1905) will bring the
proportion up to that of 1860. These statistics rest on
uncertain estimates, and should only be used for very rough
purposes.

O 2

APPENDIX I

PART I

Exercises on Chapter II

[References are to tables in the preceding pages, or to the
Statistical Abstract for the United Kingdom, 1909, Cd. 4805,
price Is. 8d.]

1. Write down the number of bushels (p. 6) in the forms
(a) to (/).

2. Add together 75,324, 79,476, 432,132, the numbers
being correct to 1 ^, 2 ^, 3 % respectively.

3. The population of a colony consists of 73,243 Euro-
peans, 7,8'* Indians and 432"°'' negroes. What is the whole
population ?

4. The average wage of 2,456°'"' workmen is 26s. 6d. (to
nearest 6d.). What is their aggregate wage ?

5. The total income of 3,254,60° persons is £243 x 10«.
What is the average income ?

6. The productivity of 4,325°°° acres is between 38 and
39 quarters of wheat per acre. A quarter weighs between
470 and 490 lbs. What is the yield in tons ?

7. £87,547 is to be raised in rates, where the assessed
annual value is £943,650. Find the least rate necessary
(fractions of one farthing not being used) and the excess
collected.

8. The quantity of wheat imported in 1894 was 70,126°°°

197

198 APPENDIX I

cwts., in 1908 91,131'""' cwts. Express the ratio in the
notations of pp. 11, 12.

9. If wages per hour rose 20 %, and the number of hours
worked per week fell 10 %, find the change in weekly wages.

10. Wages were raised 10 %, lowered 15 %, raised 20 %,
lowered 25 %, and raised 10 % in certain years, each per-
centage being reckoned on the wages current when the
change was made. Find the change in the whole period,

11. Express the 4 summary totals of the value of im-
ports {Stat. Abs., end of Table 41, p. 135) in 1894, 1900, 1908
as percentages of the grand total.

12. In the same table express the totals A to N under
III (Articles Wholly or Mainly Manufactured) as percentages
of the total of III in 1908.

[For further treatment of Approximations and the Con-
tracted Method, and numerous examples, the student is
referred to Arithinetic, Mainly Examples, G. W. Palmer,
Macmillan & Co.J

Exercises on Chapter III

1. Apply the monthly average prices of wheat in 1908
{Slat. Abs., Table 72, p. 287) to the quantities (on p. 289), to
find the total values sold as accurately as the data allow.

2. Find the average production per acre of the 6 corn
crops shown in Tables 74 and 75 for (1) Great Britain,
(2) Ireland, for the years 1894 and 1908.

Find the average yield in A, B, C together by the methods
of p. 18.

Average yield

Acreage.

per acre.

3,456,789

35-2 bushels

2,703,257

30-7 „

1,432,843

43-8 „

APPENDIX I 199

4. Using the methods of p. 20, check the averages shown
in the tables in Part I, Ch. VI, p. 51, in Part II, Ch. VI,
p. 149, suggesting the cause of the discrepancies (if any)
found.

5. Find the arithmetic average, the median, the quartiles
and the mode of the miners' ages given in Part I, Ch. V,
p. 36.

6. The prices of 4 kinds of wool are calculated in the
Stat. Als. (Table 60, p. 249), from data given in Tables 40
and 41. Check the calculations for the year 1908, and find
the average price for the four kinds lumped together. Ob-
serving that the data are incomplete in 1900, consider how
accurately an average for 1900 (comparable with those of
subsequent years) can be computed.

[Observe that camel's hair is 2d. to 2|rf. per lb. cheaper
than sheep's wool, and that the maximum import is less than
that shown under K, p. 106.]

7. Criticize the following averages : Table 25, p. 57.
Total spent in poor relief, 1905-6, England and Wales :
£11,675,959. Total relieved (p. 384): July 1st, 1905,
869,777 ; Jan. 1st, 1906, 926,741. Average number relieved,
898,259. Average cost, £12 19s. l\%d.

8. If the average wage of 55,000 men is 28s. Qd., and of
these the average for 30,000 is 25s., find the average for the
remainder.

9. Find d-^, d^ (the 1st and 9th " deciles "), in the table
on p. 23, so that "one-tenth of the wage-earners received
dj- or less, and one-tenth received dj- or more."

Exercises on Chapter IV

1. Write the net value of Consignments {Stat. Ahs.,
Table 35, pp. 80-81) from the 29 British Possessions for
1904 and 1908 in 6 columns as in the table in Part I,
Ch. IV, p. 27, and calculate the ratios of the totals. Make

200 APPENDIX I

two new columns also showing the values to the nearest
million £, and calculate the ratios.

Find also the relative and the absolute errors in the totals
of each of the columns and comment on the results.

2. Write the table of monthly prices (Stat. Ahs., Table 72,
pp. 286, 287) for the years 1894, 1900, 1908), (1) omitting
pence, (2) to the nearest shilling. Find the averages for the
years, and express them as percentages of the average found
for 1894. Comment on the result.

3. Re-write the table showing the percentage of un-
employed monthly from 1897 to 1906 (Part II, Ch. VII,
p. 158), (1) omitting the decimals, (2) to the nearest whole
numbers. Calculate the yearly means and the 10 years'
monthly averages. Comment on the result.

4. Find the total yardage and total value of woollen tissues
exported in 1908 from Stat. Ahs., Tables 48 and 49. Calculate
the average price.

Now write down the average prices of the eight qualities
enumerated from Table 61. " Weight " these averages (1)
with the number of million yards of each kind, (2) with
the values to the nearest £100,000, and find the " weighted
average " price. Explain why the result of (1) agrees very
closely with average already calculated, while the result of
(2) is 40% too large.

Exercises on Chapter V

1. Make diagrams of the type on p. 37 of the wages shown
in Part II, Ch. VI, p. 149.

2. Make circular diagrams (as on p. 47) of the main
items of revenue and expenditure in the first and last years
shown in Part II, Ch. X, p. 189.

3. From the Tables of Imports and Exports (Stat. Ahs.,
pp. 74 to 77) make the following diagrams, for the years
1894-1908 :—

APPENDIX I 201

(i) Of Imports from Foreign Countries, British Posses-
sions and Total.

(ii) Of Exports to Foreign Countries, British Posses-
sions and Total.

(iii) Total Imports and Exports.

(iv) Imports and Exports to and from British Posses-
sions.

4. From Tables 48 and 49 (Stat. Abs., pp. 162-3, 180-1)
make three diagrams of the value and quantity of Cotton
Piece Goods exported, (1) representing 100 yds. and £1 by
same unit, (2) making the lines start together, (3) making
the lines end together.

5. Represent the numbers in the table on p. 180, Pai t II,
Ch. IX, by a diagram.

6. Treat one or more of the columns of the prices in the
table on p. 128, Part II, Cli. IV, by the method of p. 42.

Exercises on Chapter VI

[Use round numbers throughout and pay attention to
clearness of meaning and legibility.]

1. Make a table of Imports and Exports (Stai. Abs.,
Table 34), grouping together the Foreign Countries in
Europe, Asia, Africa and America, and showing British
Possessions in 5 groups, from 1894 to 1905.

2. Make a similar table of Consignments (total values)
from various groups of countries, 1904 to 1908 (Stat. Abs.,
Table 35).

3. Make a table comparing Imports from and Consign-
ments from those countries in which the values are markedly
different.

4. Make a series of brief tables showing the information
as to the Woollen and Worsted industry contained in the
Statistical Abstract.

5. Make a table combining imports and exports of bullion

202 APPENDIX I

and specie with those of merchandise {Stat. Ahs., Tables 35,
36, 55 and 56) for each of the principal gold or silver
producing countries.

Exercises on Chapter. VII

1. Find by sampling the number of words (1) in a full
line of this book, (2) in lines including those at the begin-
ning and end of a paragraph. Hence estimate the number
of words in a page containing 37 lines. Calculate the pre-
cision of your estimate, and verify it by counting the number
of words in a number of pages.

2. Make a similar estimate for the average number of
letters in a word. Also by taking, say, 1000 words, find the
frequency of words of different lengths. Estimate the pre-
cision of your results, and verify it by tabulating a large
number of consecutive words.

3. Find the ratio of the number of commas to the number
of full stops in this or any other book.

4. Find whether the digits to 9 are uniformly distributed
through the table at the beginning of Chap. IV, Part I.

Exercises on Chapter VIII
1. Apply the rules of criticism given to —

(1) the various statistics relating to Paupers in Table
146, 147, pp. 384-5, Stat. Ahs. (England and Wales).

(2) the Post Office statistics, pp. 354 sqq.

(3) the categories of expenditure by Local Authorities in
the United Kingdom, Table 23, p. 55, with reference
also to Table 25.

(4) the Income Tax categories, Table 17.

APPENDIX I 203

2. How far can the statistics of (i) wages, (2) consumption
of meat, (3) value of exported manufacturers, be regarded as
tests of National Progress ?

3. Which of the average prices, Table 60, satisfy least the
criteria of paragraph 5 ?

4. Do the tables relating to railways, pp. 310-317, give the
information that statisticians want ?

Exercises on Chapter IX

1. Verify the averages in the table in paragraph 8. Deduce
the number of miles of track and of route. Show that ton-
miles per engine-hour, divided by wagon-miles per engine-
hour would equal the average fall-and-empty wagon load ;
and that wagon-miles per engine-hour, divided by train-load
would give train-miles per engine- (train and shunting) hour.

2. Consider what data would give the best information for
any business or institution with which you are acquainted.

3. Make a blank card suitable for entering details as to a
workman applying at a Labour Exchange.

4. Draw up a blank schedule suitable for tabulating details
of working-class expenditure.

5. Kequired to describe the housing accommodation of a
district. How would you proceed and what blank forms
would you use (1) if you had legal power of entry and measure-
ment, (2) if the inquiry was on a voluntary basis ?

PART II

Exercises on Chapters I and II

1. Calculate some of the birth-rates in Table 115 (Stat.
Abs., p. 362), from the number of births and from the
population stated in the preceding table.

204

APPENDIX I

2. Estimate the population of Scotland for each year from
1891 to 1901, using only the data of Table 113 (p. 361), and
compare your results with those of Table 114.

3. Work out from the Census Report for your county the
density of population in as much detail as possible in your
neighbourhood.

4. From the statistics of population, births and deaths
(Tables 113, 114, 115), find the excess of emigrants over
immigrants for Scotland and for Ireland between 1891 and
1901.

5. With the help of a diagram estimate the actual and
relative number of men between the ages 32 and 38 in
table on p. 98 above ; and also the number of children
between 7 and 14. (The whole population is given on p. 89.)

6. Find the actual numbers in various occupations from
the per mille table on p. 91, and state in what cases the
absolute numbers have increased while the relative numbers
have diminished.

7. How is it that the infant mortality rate is lower than
the death-rate between and 1 years ?

8. In Table 115 {Stat. Ahs., p. 362) calculate the population
in 1901 from the number of births and the birth-rate, as
accurately as these data allow, and compare with the two
preceding tables.

9. Find the corrected death-rate for District B to compare
with District A by both the methods described on pages
105, 106 from the following data. Find also the general
uncorrected death-rates.

District A. Relative number of persons
Death-rates .
B. Relative number of persons
Death-rates .

Years

Tears

0-5.

6-16.

15-55.

114

110

670

136

125

619

Year.s
56-

106

120

APPENDIX I 205

Exercises on Chapter III

1. Make the tables corresponding to those in paragraphs 3
and 4 for the years 1900 to 1908.

2. Make a table of the excess of imports over exports
(including bullion) for the years 1894-1908, and express
this excess year by year as a percentage of the total of
imports and exports. On the same diagram show the
numbers in this table, and the total tonnage of vessels
registered as belonging to the United Kingdom {Stat. Als.,
Table 68, p. 282).

3. Draw diagrams showing (1) total value of imports and
total tonnage of ships entered with cargoes, and (2) total
value of exports and total tonnage of ships cleared with
cargoes {Stat. Als., Table 67, pp. 280-1).

4. Draw smoothed diagrams (as Diagram III, p. 42, above)
representing the table of external trade on p. 118 above.

5. Illustrate the process described in parapraph 8 by
valuing the quantities of imported meat 1894 to 1908
{Stat. Ahs., Table 40, pp. 100, 101) at the prices of 1894
(Table 60, p. 242), and comparing the totals obtained with
the declared values (Table 41, pp. 116, 117). Show the
result on a diagram, (See also Exercise 1 on Chapter IV.)
[Work only to three significant figures.]

6. Express the following categories of exports : Food, drink
and tobacco — Coal — Other unmanufactured commodities —
Iron and steel manufactures — Cotton and wool products —
Other manufactured commodities — as percentages of total
exports of home produce, year by year from 1894 to 1908.
[Use either the table in paragraph 9, or the Stat. Ahs.]

7. Draw a diagram illustrating the increase of steamships
relative to sailing from Stat. Als., Table 67, pp. 280-1.

206 APPENDIX I

Exercises on Chapter IV

1. Make index-numbers of the prices of imported meat
from the figures obtained in Exercise V on last chapter.

2. Calculate index-numbers for 1870, 1880, 1890, 1900
aud 1908, for the eight commodities together (wheat to
coal) shown in paragraph 4 (1) taking 1865-9 as the
basis, (2) taking 1873 as the basis, (3) taking 1900 as the
basis. In each case re-write the index-numbers so that
the number for 1908 is 100. Comment on the differences
shown.

[Note.— So few commodities are, of course, insufficient for
establishing a general index-number.]

3. Transfer Sauerbeck's index-numbers from gold values
(in which they are given on p. 128) to silver values.

4. From the Stat. Ahs. (Tables 60, 61, 82) make a table
and diagram comparing the prices of pig-iron imported,
exported and produced.

Exercises on Chapter V

1. Make a table for 1900-1908 from the Stat. Ahs.
showing the value of imported raw cotton (less re-exports)
as compared with the value of exported cotton goods.
Assuming that 80% of imported cotton is used for the
foreign trade, find the value added by manufacture year
by year.

2. The Census of Production shows that the value of
the Output of Cotton Factories in 1907 was £47,000,000
more than that of cost of materials used. The value of
cotton imported and retained that year was £61,000,000,
and of exported cotton manufactures was £95,000,000. If
these statements are consistent with the 80 % assumption of
the last exercise, deduce the value of materials used (coal,
etc.) other than raw cotton.

APPENDIX I 207

3. Tabulate the values of the home production, imports
and exports, of iron, steel and their products for 1908 from
Tables 41, 49, 62, and 82 of the Stat. Ahs.

Exercises on Chapter VI

1. The wages of 2,000 men were decreased ^d. per hour
and the normal week was increased 3 hours. If before the
change the rate was 10^. and the week 50 hours, com-
pute the effect that would be shown in a " change of wages "
table.

2. If average weekly wages in Textiles, Agriculture,
Building, and Engineering had been respectively 15s., 13s.,
25s. and 27s. ; and the relative numbers employed 5, 10, 2
and 3 in 1880, compute the change per cent, for the 4 groups
together in 1890, 1900 and 1908 from the index-numbers
in paragraph 9, (1) assuming no change in the relative
numbers, (2) assuming that the numbers changed gradually
till in 1908 they were 5, 7, 3, 5.

3. If average wages rise 20%, and the retail purchasing
power of money rises 10 %, how much do average real wages
rise?

Wages.
Grade.

26s.-28s.

Number of
Year 1.

men.
Year 2.

28s.-30s.

30s.-32s.
32s.-34s.

25
25

35
25

Find the maximum and minimum change possible in
average wages consistent with promotions as shown in this
table, assuming that no man's wage was reduced,

5. Compute the lines for lads and girls on page 149 on
the assumption that all receiving less than 5s. were half-
timers (none earning less than 2s. 6d.) and supposing each

208 APPENDIX I

pair of half-timers replaced by one full-timer at their joint
wages.

Exercises on Chapter VII

1. Make diagrams illustrating the table on p. 156.

2. For lines A and B of the same table take decennial
averages for 50 periods beginning 1851, 1852, to 1900, and
represent the result in a diagram. Comment on the result.

3. Compute column D counting B2 as twice as important
at Bj.

4. Which of the lines in the table on p. 157 can be
calculated from the other lines ?

5. Make a diagram showing the general percentage
unemployment, as shown in paragraphs 6 and 7, pp. 158-160,
for every month from January 1897 to October 1909.

If seasonal changes are eliminated, which was the worst
month in 1904-5 ?

Exercises on Chapter VIII

1. Express the expenditures shown in the table on
p. 164 as percentages of the total expenditure.

2. What information does the Stat, Abs. contain as to
working-class savings ?

3. Check the percentages shown on pp. 168, 169.

4. How far can the higher expenditure shown (pp. 168-9)
for English incomes than for French or German over 40s.
be accounted for by the larger number of children ?

Exercises on Chapter IX

1. If the income-tax is Is. 2d. on unearned incomes and
9d. on earned when the total is less than £2,000, and abate-

APPENDIX I

209

ments are as shown on p. 175, find the income-tax payable
and the actual rate in the £ in the following cases —

Earned.

Unearned

Income of A . . . .300

B . . . .400

100

C . . . .300

200

D . . . .501

200

E . . . .495

200

2. The net profits of a firm were in 1900

5,500

5,000

4,500

4,400

4,000

5,000

6,500

6,000

How would these appear in the returns of income if the
average for the three preceding years were taken yearly as
described on p. 178.

Exercises on Chapter X

1. If the whale of indirect taxation were borne by working-
class families and others with incomes below £160, and the
whole of direct taxation by income-tax payers, and if the two
classes consisted respectively of 7,000,000, and 1,000,000
families and their aggregate incomes of £1,000 mln. and
£800 mln., calculate the burden per family in each case and
the proportion of taxes to income in each case. [Omit Post
Office and Crown Lands, etc.]

210 APPENDIX I

2. Estimate the aggregate income of Great Britain on the
hypothesis that among persons where the rent is
less than £25, the average family income is 8 times the rent
£25 to £50 „ „ 10

£50 to £80 „ „ 12

£80 to £500 „ „ 15

£500 and over „ „ 20

APPENDIX II

Selected List of Books of Reference

Jevons — Investigations in Currency and Finance.
QiPFEN — Economic Inquiries and Studies.
GoscnEN — Essays and Addresses on Economic Questions.
The Journal of the Royal Statistical Society. 5s. quarterly.

Government Publications.

Government publications in this list are either " Command Papers "
or " House of Commons Reports or Papers." The former are numbered
consecutively Cd. 1, Cd. 2, etc., from the year 1900 ; prior to 1900 they
were numbered C. 1, C. 2, etc., to about C. 9500. The latter are
numbered consecutively, beginning afresh each year. To order a
publication it is sufficient to give, e. g. Cd. 1761 or House of Commons
365 of 1906, without title.

Annuals.
The reference number is for the last issued prior to December 1909.

Statistical Abstract for the United Kingdom .

Statistical Abstract for the Principal Foreign

Countries . . ....

Statistical Abstract for the British Colonies
Statistical Abstract for India ....

Statistical Abstract for the British Empire

Labour Statistics

Changes in Wages and Hours of Labour .
Registrar General's Report on Births, etc.
Report of the Local Govnt. Bd.^Eng. & Wales
Trade of the United Kingdom, Vol. I for 1908
Trade of the United Kingdom, Vol. II for 1908
p 3 211

Number.

Price.

Cd. 4805

Cd. 4265

Cd. 4415

Cd. 4837

Cd. 4486

Cd. 4413

Cd. 4713

Cd. 4347

Cd. 4786

Cd. 4687

Cd. 4784

212 APPENDIX II

Number. Price.

Trade of the United Kingdom, Supplement,

1907 Cd. 4266 3 10

Trade Accounts. The 12 Months ending

December in each year . . . H.ofC. 25xi. of 1908 1 2

Navigation and Shipping, 1 908 Cd. 4789 2 8

Beports of Commissioners of the Inland

Revenue . . . . Cd. 4868 1 7

Special RepoHs.

Year.

Statistical Tables and Charts relating to British

and Foreign Trade and Industry . . 1903 Cd. 1761 3 6
Statistical Tables and Charts relating to British

and Foreign Trade and Industry . . 1904 Cd. 2337 3 6
Statistical Tables and Charts relating to British

and Foreign Trade and Industry
Public Health and Social Conditions .
Decennial Supplement to Registrar-General's

Report, Part I

Decennial Supplement to Registrar-General's

Report, Part II.

Report of Committee on the Census
Report of Committee on the Income-tax
Population Census — England and Wales :

Summary Tables, 1901 ....
Population Census— England and Wales :

General Report, 1901

Wholesale and Retail Prices ....
Census of Production : Preliminary Tables .
Wage Census. I. Textiles .
Wage Census. II. Clothing Trades

Standard Time Rates

Standard Piece Rates

Rents, Prices, Wages, etc., in Towns in the

United Kingdom

Rents, Prices, Wages, etc., in Towns in

Germany 1908

Rents, Prices, Wages, etc., in Towns in France

1909

Cd. 4954

1909

Cd. 4671

1907

Cd. 2618

1908

Cd. 2619

1890

C. 6071

1906 il.ofC. 365

1903

Cd. 1523

1904

Cd. 2174

1903 H.ofC. 321

1909

Cd. 4896

1903

Ca. 4545

1909
fl894
-^1906
1.1909
/1894
11900

Cd. 4844
C. 7567 ii.
Cd. 3245
Cd. 4924
C. 7567 i.
Cd. 144

2
1

5
3

7
6

1908

Cd. 3864

1908

Cd. 4032

1909

Cd. 4512

INDEX

Abatements, 173, 175

Accuracy, 6-13, 69, 109

Addition, approximate, 8

Administrative counties, 87, 99

Ages, 36, 46, 98

Agricultural statistics, 134; wages, 147

Ancient counties, 99, 100

Approximation, 6-13

Argentina, trade of, 116

Arithmetical average, 17 ; errors of,
31-33

Attack rate, 108

Australia, trade of, 116

Austria, trade of, 116, 122

Averages, 15-28, 54, 67, 108, 110;
accuracy of, 27-34 ; calculation of,
19, 20 ; descriptive, 20 ; errors of,
29-31 ; in sampling, 58 ; weighted,
17

Belgium, trade of, 116, 122

Bertillon, M., 102

Biassed errors, 29, 31, 32

Birth-rate, 2, 67, 102 sqq.

Blank fonns, 81

Boroughs, 87, 100, 101

Brazil, trade of, 116

Budgets, working-class, 58, 166-170

Building, numbers, 91 ; unemploy-
ment, 153, 158 ; wages, 143, 147

Bulgaria, trade of, 116

Bullion, imports and exports of,
113, 114, 116, 117

Canada, trade of, 116
Capital, 182-4
Case fatality rate, 108
Census, population, 85-101 ; produc-
tion, 133 ; wages, 142
Changes in wages, 141-8
Chapman, Prof ,,152
Chile, trade of, 116
China, trade of, 116
Civil parishes, 87, 99, 101

Classes, statistical, 1

Clothing trade, wages, 149

Coal, exports, 121 ; prices, 128-9 ;
production, 133

Coal miners, numbers, 91 ; wages,
143-5

Collection of statistics, 80

Comparison, criticism of, 68 ; error
in, 32 ; nature of, 11

Co-operative societies, 166

Corrected death-rates, 105-6

Correction factor, 105-6

Costa Rica, trade of, 116

Cotton, exports, 121 ; numbers era-
ployed, 92 ; prices, 1 28-9 ; pro-
duction, 133 ; wages, 22, 149

Counties, 87, 99, 100

County boroughs, 87, 100

Craigie, Major, 134

Crawford, R. F., 134

Criticism of statistics, 64-70

Customs, 185-6, 188-9

Death-rates, 2, 65, 102 sqq.

Deciles, 62, 199

Definition, importance of, 3, 57, 64,
108, 132

Denmark, trade of, 116

Density of population, 92-3

Diagrams, 22, 35-49, 54, 109; cir-
cular, 47 ; cumulative, 37

Division, approximate, 10

Dudfield, Dr., 102

Earned income, 181
Ecclesiastical parishes, 87, 101
Eamomist, The, 126
Education, expenditure on, 186-7,

194
Emigration, 95, 96
Employment, 151-160
Engineering, unemployment, 153,

156, 157-9 ; wages, 143, 147

213

214

INDEX

England and Wales, census of, 86 ;

population, 89
Errors, absolute, 28-34 ; biassed, 29 ;

meaning and nature of, 7, 8, 28 ;

percentage, 28 ; relative, 28-32 ;

unbiassed, 29
Estate duty statistics, 182-5, 189
Excise, 185, 188, 189
Exemptions, 173, 174, 177,180
Exports, 111-122 ; iron and steel,

45 ; prices, 126-8, 132 ; worsted

goods, 76

Finland, trade of, 116

Fish, 134

Fluctuations, 41-3, 69

Food, imports and exports, 121

Foreign trade, 111-122

France, trade of, 116, 122 ; shipping

of, 123
Frequency curves, 22
Friendly societies, 165

Geemany, trade of, 116, 122 ; ship-
ping of, 123
Greece, trade of, 116
Groups, statistical, 1, 35, 36

Hallswokth, H. M., 152
Hayward, T. E., 97
Holland, trade of, 116, 122
Homogeneity, 65, 67, 75, 79, 90
Hooker, E. H., 134
HuU, 88

Immigbation, 95, 96
Imports, 119-122 ; prices,126-31,132
Income-tax statistics, 173-9, 184, 189
Index-numbers, prices, 76, 128-132 ;

wages, 146-7, 180 ; incomes, 180
India, trade of, 116
Indices, 66

Infant mortality, 104, 105, 107
Inhabited house duty, 190-2
Inland Eeveniie, 186-6
International trade, 116
Ireland, census, 86 ; population, 89
Iron and steel, exports, 45, 121 ;

prices, 128-9 ; production, 133,

135-6 ; manufacture, wages in, 143-

145
Italy, trade of, 116 ; shipping of,

123

Japan, trade of, 116; shipping of, 123
Jute prices, 128-9

Labour Gazette, 126, 142, 152-3
London, 27, 89, 101
Longstaff, Dr., 110

Mallet, B., 184
Marriage-rates, 2, 102, 104
Mean, 21, 22

Median, 22, 24, 33, 39, 62
Medical officers of health, 102, 107
Merchant navies, 123
Metropolitan boroughs, 27, 101
Mexico, trade of, 116
Middlesborough, 93
Mineral statistics, 134
Miners' wages, 143-5
Mixture for sampling, 66
Mode, 21, 22, 32, 39
Morbidity rates, 108
Multiplication, approximate, 9
Municipal boroughs, 87, 100

National income, 171-184
Natural increase, 95, 97
Newsholme, Dr., 102
Normal week, 137

Norwiiy, trade of, 116 ; shipping of,
123

Oooitpations, numbers in, 90-2
Overcrowding, 80, 81, 94

Pakishes, 87, 99

Percentages, 67 ; mistakes in, 14 ;
method of, 11

Peru, trade of, 116

Piece-rates, 138-9

Poor relief, expenditure on, 194

Population, 6-7 ; census, 85-101 ;
England and Wales, 89, 95 ; Ire-
land, 89 ; London, 27, 89 ; Scot-
land, 89 ; United Kingdom, 89, 96

Portugal, trade of, 116

Predominant rate, 21

Prices, 126-132

Printers, unemployment, 153, 158

Production, 133-6

Published statistics, rules for using,
64-70

QuAKTiLES, 24, 39, 62

Railway statistics, 72-4, 124
Random selection, 57
Bates, as averages, 16, 67

,, see birth-, death-, marriage-
Rates, statistics of, 193-5

INDEX

215

Ratio, nature of, 11 ; error in, 32

Eaw material, imports and exports,
121

Real wages, 148, 150

Registrar-general, 86, 102 sqq.

Registration, counties, 87, 99 ; dis-
tricts, 87, 99

Retail prices, 127

Rew, R. H., 134

Rural districts, 87, 89, 94, 100, 101

Rural population, 89

Russia, trade of, 116, 122

Sampling, 56-63

Sauerbeck, A., indez-numbers com-
puted by, 126, 128, 130, 131

Savings, working-class, 166

Sohloss, D. F., 139

Scotland, census, 86 ; population, 89

Series, statistical, 2, 35, 41, 43, 69

Shipbuilding, unemployment, 153,
156, 158-9

Shipping, 122-5

Shipping tonnage, note on, 124-5

Ships, built, 135 ; exported, 121

Silver, prices, 128-9

Sliding-scales, 141 [note)

Smoothing, 41, 96

South Africa, trade of, 116

Spain, trade of, 116 ; shipping of,
123

Square root, approximate, 10

Standard death-rate, 105-6

Statist, The, 126

Strikes, statistics of, 160-3

Subtraction, approximate, 8

Sugar, prices, 128-9

Sweden, trade of, 116 : shipping of,
123

Switzerland, trade of, 116, 122

Tabulation, 50-55, 108
Tax statistics, 185-195
Tea, prices, 128-9

Textile manufactures, numbers occu-
pied, 91 ; wages, 143-5, 147
Time wages, 138, 140
Tinplate production, 133
Ton-miles, 73
Tonnage, shipping, 122-5
Trade-union statistics, 163-4
Transhipment, 113
Trend, 41-43, 69

Unbiassed errors, 29, 31, 32
Unearned income, 181-2
Unemployment, 4, 151-160
U.S.A., trade of, 115-16; shipping of,

123
Urban districts, 87, 89, 94, 100, 101
Urban population, 88, 89
Uruguay, trade of, 116

Vital statistics, 102-110

Wages, 67, 77 ; index-numbers of,
146-7 ; statistics of, 22,51, 137-150

Wages and incomes, changes com-
pared, 180

Waters, A. C, 97

Weights, weighted averages, weighted
totals, 17, 18, 32, 76, 77

Wheat, prices, 128-9

Wood, G. H., 145, 148

Wool, exports, 121 ; numbers em-
ployed in manufactm-e, 92 : prices
128 ; statistics, 136

Worsted manufactures, exports, 76

YOEKSHIBE, 93

EiCHARD Clay & Sons, Limited,

BREAD STREET BILL, E.G., AND
BUNGAY, SUFFOLK.

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